{"id":1769,"date":"2018-01-11T20:43:00","date_gmt":"2018-01-11T20:43:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/integrals-involving-exponential-and-logarithmic-functions\/"},"modified":"2018-01-31T20:57:48","modified_gmt":"2018-01-31T20:57:48","slug":"integrals-involving-exponential-and-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/chapter\/integrals-involving-exponential-and-logarithmic-functions\/","title":{"raw":"5.6 Integrals Involving Exponential and Logarithmic Functions","rendered":"5.6 Integrals Involving Exponential and Logarithmic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Integrate functions involving exponential functions.<\/li>\r\n \t<li>Integrate functions involving logarithmic functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1170572130673\">Exponential and logarithmic functions are used to model population growth, cell growth, and financial growth, as well as depreciation, radioactive decay, and resource consumption, to name only a few applications. In this section, we explore integration involving exponential and logarithmic functions.<\/p>\r\n\r\n<div id=\"fs-id1170572137421\" class=\"bc-section section\">\r\n<h1>Integrals of Exponential Functions<\/h1>\r\n<p id=\"fs-id1170571602573\">The exponential function is perhaps the most efficient function in terms of the operations of calculus. The exponential function, [latex]y={e}^{x},[\/latex] is its own derivative and its own integral.<\/p>\r\n\r\n<div id=\"fs-id1170572370390\" class=\"textbox key-takeaways\">\r\n<h3>Rule: Integrals of Exponential Functions<\/h3>\r\n<p id=\"fs-id1170572375243\">Exponential functions can be integrated using the following formulas.<\/p>\r\n\r\n<div id=\"fs-id1170572374424\" class=\"equation\">[latex]\\begin{array}{ccc}\\int {e}^{x}dx\\hfill &amp; =\\hfill &amp; {e}^{x}+C\\hfill \\\\ \\int {a}^{x}dx\\hfill &amp; =\\hfill &amp; \\frac{{a}^{x}}{\\text{ln}a}+C\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id11705722115180\" class=\"textbox examples\">\r\n<div id=\"fs-id1170572606041\" class=\"exercise\">\r\n<div id=\"fs-id1170572222389\" class=\"textbox\">\r\n<h3>Finding an Antiderivative of an Exponential Function<\/h3>\r\n<p id=\"fs-id1170572150624\">Find the antiderivative of the exponential function [latex]e[\/latex]<sup>\u2212[latex]x[\/latex]<\/sup>.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571601500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571601500\"]\r\n<p id=\"fs-id1170571601500\">Use substitution, setting [latex]u=\\text{\u2212}x,[\/latex] and then [latex]du=-1dx.[\/latex] Multiply the <em>du<\/em> equation by \u22121, so you now have [latex]\\text{\u2212}du=dx.[\/latex] Then,<\/p>\r\n\r\n<div id=\"fs-id1170572454214\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\int {e}^{\\text{\u2212}x}dx\\hfill &amp; =\\text{\u2212}\\int {e}^{u}du\\hfill \\\\ \\\\ &amp; =\\text{\u2212}{e}^{u}+C\\hfill \\\\ &amp; =\\text{\u2212}{e}^{\\text{\u2212}x}+C.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572248011\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170571655362\" class=\"exercise\">\r\n<div id=\"fs-id1170572206116\" class=\"textbox\">\r\n<p id=\"fs-id1170572430365\">Find the antiderivative of the function using substitution: [latex]{x}^{2}{e}^{-2{x}^{3}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572269592\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572269592\"]\r\n<p id=\"fs-id1170572269592\">[latex]\\int {x}^{2}{e}^{-2{x}^{3}}dx=-\\frac{1}{6}{e}^{-2{x}^{3}}+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170573205217\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170571622632\">Let [latex]u[\/latex] equal the exponent on [latex]e[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572607911\">A common mistake when dealing with exponential expressions is treating the exponent on [latex]e[\/latex] the same way we treat exponents in polynomial expressions. We cannot use the power rule for the exponent on [latex]e[\/latex]. This can be especially confusing when we have both exponentials and polynomials in the same expression, as in the previous checkpoint. In these cases, we should always double-check to make sure we\u2019re using the right rules for the functions we\u2019re integrating.<\/p>\r\n\r\n<div id=\"fs-id1170572209244\" class=\"textbox examples\">\r\n<h3>Square Root of an Exponential Function<\/h3>\r\n<div id=\"fs-id1170571602248\" class=\"exercise\">\r\n<div id=\"fs-id1170572509298\" class=\"textbox\">\r\n<p id=\"fs-id1170571773419\">Find the antiderivative of the exponential function [latex]{e}^{x}\\sqrt{1+{e}^{x}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572551693\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572551693\"]\r\n<p id=\"fs-id1170572551693\">First rewrite the problem using a rational exponent:<\/p>\r\n\r\n<div id=\"fs-id1170572549246\" class=\"equation unnumbered\">[latex]\\int {e}^{x}\\sqrt{1+{e}^{x}}dx=\\int {e}^{x}{(1+{e}^{x})}^{1\\text{\/}2}dx.[\/latex]<\/div>\r\nUsing substitution, choose [latex]u=1+{e}^{x}.u=1+{e}^{x}.[\/latex] Then, [latex]du={e}^{x}dx.[\/latex] We have (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_05_06_001\">(Figure)<\/a>)\r\n<div id=\"fs-id1170572373474\" class=\"equation unnumbered\">[latex]\\int {e}^{x}{(1+{e}^{x})}^{1\\text{\/}2}dx=\\int {u}^{1\\text{\/}2}du.[\/latex]<\/div>\r\n<p id=\"fs-id1170572307548\">Then<\/p>\r\n\r\n<div id=\"fs-id1170572560605\" class=\"equation unnumbered\">[latex]\\int {u}^{1\\text{\/}2}du=\\frac{{u}^{3\\text{\/}2}}{3\\text{\/}2}+C=\\frac{2}{3}{u}^{3\\text{\/}2}+C=\\frac{2}{3}{(1+{e}^{x})}^{3\\text{\/}2}+C.[\/latex]<\/div>\r\n<div id=\"CNX_Calc_Figure_05_06_001\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204252\/CNX_Calc_Figure_05_06_001.jpg\" alt=\"A graph of the function f(x) = e^x * sqrt(1 + e^x), which is an increasing concave up curve, over [-3, 1]. It begins close to the x axis in quadrant two, crosses the y axis at (0, sqrt(2)), and continues to increase rapidly.\" width=\"325\" height=\"208\" \/> Figure 1. The graph shows an exponential function times the square root of an exponential function.[\/caption]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572207056\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572451630\" class=\"exercise\">\r\n<div id=\"fs-id1170571609364\" class=\"textbox\">\r\n<p id=\"fs-id1170572363371\">Find the antiderivative of [latex]{e}^{x}{(3{e}^{x}-2)}^{2}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<p id=\"fs-id1170572209127\">[latex]\\int {e}^{x}{(3{e}^{x}-2)}^{2}dx=\\frac{1}{9}{(3{e}^{x}-2)}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170573502097\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\nLet [latex]u=3{e}^{x}-2u=3{e}^{x}-2.[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<div id=\"fs-id1170572240614\" class=\"exercise\">\r\n<div id=\"fs-id1170572209184\" class=\"textbox\">\r\n<h3>Using Substitution with an Exponential Function<\/h3>\r\nUse substitution to evaluate the indefinite integral [latex]\\int 3{x}^{2}{e}^{2{x}^{3}}dx.[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572216534\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572216534\"]\r\n<p id=\"fs-id1170572216534\">Here we choose to let [latex]u[\/latex] equal the expression in the exponent on [latex]e[\/latex]. Let [latex]u=2{x}^{3}[\/latex] and [latex]du=6{x}^{2}dx..[\/latex] Again, <em>du<\/em> is off by a constant multiplier; the original function contains a factor of 3[latex]x[\/latex]<sup>2<\/sup>, not 6[latex]x[\/latex]<sup>2<\/sup>. Multiply both sides of the equation by [latex]\\frac{1}{2}[\/latex] so that the integrand in [latex]u[\/latex] equals the integrand in [latex]x[\/latex]. Thus,<\/p>\r\n\r\n<div id=\"fs-id1170572470447\" class=\"equation unnumbered\">[latex]\\int 3{x}^{2}{e}^{2{x}^{3}}dx=\\frac{1}{2}\\int {e}^{u}du.[\/latex]<\/div>\r\n<p id=\"fs-id1170572101855\">Integrate the expression in [latex]u[\/latex] and then substitute the original expression in [latex]x[\/latex] back into the [latex]u[\/latex] integral:<\/p>\r\n\r\n<div id=\"fs-id1170572448333\" class=\"equation unnumbered\">[latex]\\frac{1}{2}\\int {e}^{u}du=\\frac{1}{2}{e}^{u}+C=\\frac{1}{2}{e}^{2{x}^{3}}+C.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572551647\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572294922\" class=\"exercise\">\r\n<div id=\"fs-id1170572095031\" class=\"textbox\">\r\n<p id=\"fs-id1170572295300\">Evaluate the indefinite integral [latex]\\int 2{x}^{3}{e}^{{x}^{4}}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572242344\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572242344\"]\r\n<p id=\"fs-id1170572242344\">[latex]\\int 2{x}^{3}{e}^{{x}^{4}}dx=\\frac{1}{2}{e}^{{x}^{4}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170573414530\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170572232621\">Let [latex]u={x}^{4}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170571733816\">As mentioned at the beginning of this section, exponential functions are used in many real-life applications. The number [latex]e[\/latex] is often associated with compounded or accelerating growth, as we have seen in earlier sections about the derivative. Although the derivative represents a rate of change or a growth rate, the integral represents the total change or the total growth. Let\u2019s look at an example in which integration of an exponential function solves a common business application.<\/p>\r\n<p id=\"fs-id1170571571226\">A <span class=\"no-emphasis\">price\u2013demand function<\/span> tells us the relationship between the quantity of a product demanded and the price of the product. In general, price decreases as quantity demanded increases. The marginal price\u2013demand function is the derivative of the price\u2013demand function and it tells us how fast the price changes at a given level of production. These functions are used in business to determine the price\u2013elasticity of demand, and to help companies determine whether changing production levels would be profitable.<\/p>\r\n\r\n<div id=\"fs-id1170572100406\" class=\"textbox examples\">\r\n<div id=\"fs-id1170571597871\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n<h3>Finding a Price\u2013Demand Equation<\/h3>\r\n<p id=\"fs-id1170571526652\">Find the price\u2013demand equation for a particular brand of toothpaste at a supermarket chain when the demand is 50 tubes per week at $2.35 per tube, given that the marginal price\u2014demand function, [latex]{p}^{\\prime }(x),[\/latex] for [latex]x[\/latex] number of tubes per week, is given as<\/p>\r\n\r\n<div id=\"fs-id1170572141865\" class=\"equation unnumbered\">[latex]p\\text{'}(x)=-0.015{e}^{-0.01x}.[\/latex]<\/div>\r\n<p id=\"fs-id1170571595414\">If the supermarket chain sells 100 tubes per week, what price should it set?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571543163\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571543163\"]\r\n<p id=\"fs-id1170571543163\">To find the price\u2013demand equation, integrate the marginal price\u2013demand function. First find the antiderivative, then look at the particulars. Thus,<\/p>\r\n\r\n<div id=\"fs-id1170572216339\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ p(x)\\hfill &amp; =\\int -0.015{e}^{-0.01x}dx\\hfill \\\\ &amp; =-0.015\\int {e}^{-0.01x}dx.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572507579\">Using substitution, let [latex]u=-0.01x[\/latex] and [latex]du=-0.01dx.[\/latex] Then, divide both sides of the <em>du<\/em> equation by \u22120.01. This gives<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\frac{-0.015}{-0.01}\\int {e}^{u}du\\hfill &amp; =1.5\\int {e}^{u}du\\hfill \\\\ \\\\ &amp; =1.5{e}^{u}+C\\hfill \\\\ &amp; =1.5{e}^{-0.01x}+C.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572111467\">The next step is to solve for <em>C<\/em>. We know that when the price is $2.35 per tube, the demand is 50 tubes per week. This means<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ p(50)\\hfill &amp; =1.5{e}^{-0.01(50)}+C\\hfill \\\\ &amp; =2.35.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170571569187\">Now, just solve for <em>C<\/em>:<\/p>\r\n\r\n<div id=\"fs-id1170572415392\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ C\\hfill &amp; =2.35-1.5{e}^{-0.5}\\hfill \\\\ &amp; =2.35-0.91\\hfill \\\\ &amp; =1.44.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572140846\">Thus,<\/p>\r\n\r\n<div id=\"fs-id1170572375703\" class=\"equation unnumbered\">[latex]p(x)=1.5{e}^{-0.01x}+1.44.[\/latex]<\/div>\r\n<p id=\"fs-id1170571547617\">If the supermarket sells 100 tubes of toothpaste per week, the price would be<\/p>\r\n\r\n<div id=\"fs-id1170571653359\" class=\"equation unnumbered\">[latex]p(100)=1.5{e}^{-0.01(100)}+1.44=1.5{e}^{-1}+1.44\\approx 1.99.[\/latex]<\/div>\r\n<p id=\"fs-id1170572393399\">The supermarket should charge $1.99 per tube if it is selling 100 tubes per week.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571637964\" class=\"textbox examples\">\r\n<div id=\"fs-id1170572204890\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n<h3>Evaluating a Definite Integral Involving an Exponential Function<\/h3>\r\n<p id=\"fs-id1170571657942\">Evaluate the definite integral [latex]{\\int }_{1}^{2}{e}^{1-x}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572247799\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572247799\"]\r\n<p id=\"fs-id1170572247799\">Again, substitution is the method to use. Let [latex]u=1-x,[\/latex] so [latex]du=-1dx[\/latex] or [latex]\\text{\u2212}du=dx.[\/latex] Then [latex]\\int {e}^{1-x}dx=\\text{\u2212}\\int {e}^{u}du.[\/latex] Next, change the limits of integration. Using the equation [latex]u=1-x,[\/latex] we have<\/p>\r\n\r\n<div id=\"fs-id1170571652053\" class=\"equation unnumbered\">[latex]\\begin{array}{c}u=1-(1)=0\\hfill \\\\ u=1-(2)=-1.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170571645662\">The integral then becomes<\/p>\r\n\r\n<div id=\"fs-id1170571645665\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}{\\int }_{1}^{2}{e}^{1-x}dx\\hfill &amp; =\\text{\u2212}{\\int }_{0}^{-1}{e}^{u}du\\hfill \\\\ \\\\ \\\\ &amp; ={\\int }_{-1}^{0}{e}^{u}du\\hfill \\\\ &amp; ={{e}^{u}|}_{-1}^{0}\\hfill \\\\ &amp; ={e}^{0}-({e}^{-1})\\hfill \\\\ &amp; =\\text{\u2212}{e}^{-1}+1.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572495875\">See <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_05_06_002\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_05_06_002\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204254\/CNX_Calc_Figure_05_06_002.jpg\" alt=\"A graph of the function f(x) = e^(1-x) over [0, 3]. It crosses the y axis at (0, e) as a decreasing concave up curve and symptotically approaches 0 as x goes to infinity.\" width=\"325\" height=\"208\" \/> Figure 2. The indicated area can be calculated by evaluating a definite integral using substitution.[\/caption]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571599287\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170571599290\" class=\"exercise\">\r\n<div id=\"fs-id1170571599292\" class=\"textbox\">\r\n<p id=\"fs-id1170571599295\">Evaluate [latex]{\\int }_{0}^{2}{e}^{2x}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572274760\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572274760\"]\r\n<p id=\"fs-id1170572274760\">[latex]\\frac{1}{2}{\\int }_{0}^{4}{e}^{u}du=\\frac{1}{2}({e}^{4}-1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170573206282\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170572242289\">Let [latex]u=2x.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572624841\" class=\"textbox examples\">\r\n<h3>Growth of Bacteria in a Culture<\/h3>\r\n<div id=\"fs-id1170572624843\" class=\"exercise\">\r\n<div id=\"fs-id1170572337835\" class=\"textbox\">\r\n<p id=\"fs-id1170572337840\">Suppose the rate of <span class=\"no-emphasis\">growth of bacteria<\/span> in a Petri dish is given by [latex]q(t)={3}^{t},[\/latex] where [latex]t[\/latex] is given in hours and [latex]q(t)[\/latex] is given in thousands of bacteria per hour. If a culture starts with 10,000 bacteria, find a function [latex]Q(t)[\/latex] that gives the number of bacteria in the Petri dish at any time [latex]t[\/latex]. How many bacteria are in the dish after 2 hours?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571699005\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571699005\"]\r\n<p id=\"fs-id1170571699005\">We have<\/p>\r\n\r\n<div id=\"fs-id1170571699008\" class=\"equation unnumbered\">[latex]Q(t)=\\int {3}^{t}dt=\\frac{{3}^{t}}{\\text{ln}3}+C.[\/latex]<\/div>\r\n<p id=\"fs-id1170571613514\">Then, at [latex]t=0[\/latex] we have [latex]Q(0)=10=\\frac{1}{\\text{ln}3}+C,[\/latex] so [latex]C\\approx 9.090[\/latex] and we get<\/p>\r\n\r\n<div id=\"fs-id1170571775806\" class=\"equation unnumbered\">[latex]Q(t)=\\frac{{3}^{t}}{\\text{ln}3}+9.090.[\/latex]<\/div>\r\n<p id=\"fs-id1170571609258\">At time [latex]t=2,[\/latex] we have<\/p>\r\n\r\n<div id=\"fs-id1170571698223\" class=\"equation unnumbered\">[latex]Q(2)=\\frac{{3}^{2}}{\\text{ln}3}+9.090[\/latex]<\/div>\r\n<div id=\"fs-id1170571637497\" class=\"equation unnumbered\">[latex]=17.282.[\/latex]<\/div>\r\n<p id=\"fs-id1170571637508\">After 2 hours, there are 17,282 bacteria in the dish.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571637514\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170571637518\" class=\"exercise\">\r\n<div id=\"fs-id1170571637520\" class=\"textbox\">\r\n<p id=\"fs-id1170572437454\">From <a class=\"autogenerated-content\" href=\"#fs-id1170572624841\">(Figure)<\/a>, suppose the bacteria grow at a rate of [latex]q(t)={2}^{t}.[\/latex] Assume the culture still starts with 10,000 bacteria. Find [latex]Q(t).[\/latex] How many bacteria are in the dish after 3 hours?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572444236\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572444236\"]\r\n<p id=\"fs-id1170572444236\">[latex]Q(t)=\\frac{{2}^{t}}{\\text{ln}2}+8.557.[\/latex] There are 20,099 bacteria in the dish after 3 hours.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170573623952\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170572444227\">Use the procedure from <a class=\"autogenerated-content\" href=\"#fs-id1170572624841\">(Figure)<\/a> to solve the problem.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571807209\" class=\"textbox examples\">\r\n<h3>Fruit Fly Population Growth<\/h3>\r\n<div id=\"fs-id1170571807211\" class=\"exercise\">\r\n<div id=\"fs-id1170571807213\" class=\"textbox\">\r\n<p id=\"fs-id1170571807219\">Suppose a population of <span class=\"no-emphasis\">fruit flies<\/span> increases at a rate of [latex]g(t)=2{e}^{0.02t},[\/latex] in flies per day. If the initial population of fruit flies is 100 flies, how many flies are in the population after 10 days?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572183855\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572183855\"]\r\n<p id=\"fs-id1170572183855\">Let [latex]G(t)[\/latex] represent the number of flies in the population at time [latex]t[\/latex]. Applying the net change theorem, we have<\/p>\r\n\r\n<div id=\"fs-id1170571712220\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ G(10)\\hfill &amp; =G(0)+{\\int }_{0}^{10}2{e}^{0.02t}dt\\hfill \\\\ &amp; =100+{\\left[\\frac{2}{0.02}{e}^{0.02t}\\right]|}_{0}^{10}\\hfill \\\\ &amp; =100+{\\left[100{e}^{0.02t}\\right]|}_{0}^{10}\\hfill \\\\ &amp; =100+100{e}^{0.2}-100\\hfill \\\\ &amp; \\approx 122.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572338467\">There are 122 flies in the population after 10 days.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572338474\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572338477\" class=\"exercise\">\r\n<div id=\"fs-id1170572338479\" class=\"textbox\">\r\n<p id=\"fs-id1170572338481\">Suppose the rate of growth of the fly population is given by [latex]g(t)={e}^{0.01t},[\/latex] and the initial fly population is 100 flies. How many flies are in the population after 15 days?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170571653080\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571653080\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571653080\"]There are 116 flies.<\/div>\r\n<div id=\"fs-id1170573548872\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170571653073\">Use the process from <a class=\"autogenerated-content\" href=\"#fs-id1170571807209\">(Figure)<\/a> to solve the problem.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<div id=\"fs-id1170571653091\" class=\"exercise\">\r\n<div id=\"fs-id1170571653093\" class=\"textbox\">\r\n<h3>Evaluating a Definite Integral Using Substitution<\/h3>\r\n<p id=\"fs-id1170572217480\">Evaluate the definite integral using substitution: [latex]{\\int }_{1}^{2}\\frac{{e}^{1\\text{\/}x}}{{x}^{2}}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572587719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572587719\"]\r\n<p id=\"fs-id1170572587719\">This problem requires some rewriting to simplify applying the properties. First, rewrite the exponent on [latex]e[\/latex] as a power of [latex]x[\/latex], then bring the [latex]x[\/latex]<sup>2<\/sup> in the denominator up to the numerator using a negative exponent. We have<\/p>\r\n\r\n<div id=\"fs-id1170572587738\" class=\"equation unnumbered\">[latex]{\\int }_{1}^{2}\\frac{{e}^{1\\text{\/}x}}{{x}^{2}}dx={\\int }_{1}^{2}{e}^{{x}^{-1}}{x}^{-2}dx.[\/latex]<\/div>\r\n<p id=\"fs-id1170571636291\">Let [latex]u={x}^{-1},[\/latex] the exponent on [latex]e[\/latex]. Then<\/p>\r\n\r\n<div id=\"fs-id1170571636314\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill du&amp; =\\text{\u2212}{x}^{-2}dx\\hfill \\\\ \\hfill -du&amp; ={x}^{-2}dx.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170571599633\">Bringing the negative sign outside the integral sign, the problem now reads<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\text{\u2212}\\int {e}^{u}du.[\/latex]<\/div>\r\n<p id=\"fs-id1170571599663\">Next, change the limits of integration:<\/p>\r\n\r\n<div id=\"fs-id1170571599666\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ u={(1)}^{-1}=1\\hfill \\\\ u={(2)}^{-1}=\\frac{1}{2}.\\hfill \\end{array}[\/latex]<\/div>\r\nNotice that now the limits begin with the larger number, meaning we must multiply by \u22121 and interchange the limits. Thus,\r\n<div id=\"fs-id1170572293466\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ \\\\ \\text{\u2212}{\\int }_{1}^{1\\text{\/}2}{e}^{u}du\\hfill &amp; ={\\int }_{1\\text{\/}2}^{1}{e}^{u}du\\hfill \\\\ &amp; ={e}^{u}{|}_{1\\text{\/}2}^{1}\\hfill \\\\ &amp; =e-{e}^{1\\text{\/}2}\\hfill \\\\ &amp; =e-\\sqrt{e}.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572554374\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572554378\" class=\"exercise\">\r\n<div id=\"fs-id1170572554380\" class=\"textbox\">\r\n<p id=\"fs-id1170572554382\">Evaluate the definite integral using substitution: [latex]{\\int }_{1}^{2}\\frac{1}{{x}^{3}}{e}^{4{x}^{-2}}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572628419\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572628419\"]\r\n<p id=\"fs-id1170572628419\">[latex]{\\int }_{1}^{2}\\frac{1}{{x}^{3}}{e}^{4{x}^{-2}}dx=\\frac{1}{8}\\left[{e}^{4}-e\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170571419835\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170572554429\">Let [latex]u=4{x}^{-2}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572565328\" class=\"bc-section section\">\r\n<h1>Integrals Involving Logarithmic Functions<\/h1>\r\n<p id=\"fs-id1170572565333\">Integrating functions of the form [latex]f(x)={x}^{-1}[\/latex] result in the absolute value of the natural log function, as shown in the following rule. Integral formulas for other logarithmic functions, such as [latex]f(x)=\\text{ln}x[\/latex] and [latex]f(x)={\\text{log}}_{a}x,[\/latex] are also included in the rule.<\/p>\r\n\r\n<div id=\"fs-id1170572543658\" class=\"textbox key-takeaways\">\r\n<h3>Rule: Integration Formulas Involving Logarithmic Functions<\/h3>\r\n<p id=\"fs-id1170572543663\">The following formulas can be used to evaluate integrals involving logarithmic functions.<\/p>\r\n\r\n<div id=\"fs-id1170572543666\" class=\"equation\">[latex]\\begin{array}{ccc}\\hfill \\int {x}^{-1}dx&amp; =\\hfill &amp; \\text{ln}|x|+C\\hfill \\\\ \\hfill \\int \\text{ln}xdx&amp; =\\hfill &amp; x\\text{ln}x-x+C=x(\\text{ln}x-1)+C\\hfill \\\\ \\hfill \\int {\\text{log}}_{a}xdx&amp; =\\hfill &amp; \\frac{x}{\\text{ln}a}(\\text{ln}x-1)+C\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571653392\" class=\"textbox examples\">\r\n<h3>Finding an Antiderivative Involving [latex]\\text{ln}x[\/latex]<\/h3>\r\n<div id=\"fs-id1170571653394\" class=\"exercise\">\r\n<div id=\"fs-id1170571653396\" class=\"textbox\">\r\n<p id=\"fs-id1170571653410\">Find the antiderivative of the function [latex]\\frac{3}{x-10}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571653435\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571653435\"]\r\n<p id=\"fs-id1170571653435\">First factor the 3 outside the integral symbol. Then use the [latex]u[\/latex]<sup>\u22121<\/sup> rule. Thus,<\/p>\r\n\r\n<div id=\"fs-id1170571653444\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}\\int \\frac{3}{x-10}dx\\hfill &amp; =3\\int \\frac{1}{x-10}dx\\hfill \\\\ \\\\ \\\\ &amp; =3\\int \\frac{du}{u}\\hfill \\\\ &amp; =3\\text{ln}|u|+C\\hfill \\\\ &amp; =3\\text{ln}|x-10|+C,x\\ne 10.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170571597433\">See <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_05_06_003\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_05_06_003\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204256\/CNX_Calc_Figure_05_06_003.jpg\" alt=\"A graph of the function f(x) = 3 \/ (x \u2013 10). There is an asymptote at x=10. The first segment is a decreasing concave down curve that approaches 0 as x goes to negative infinity and approaches negative infinity as x goes to 10. The second segment is a decreasing concave up curve that approaches infinity as x goes to 10 and approaches 0 as x approaches infinity.\" width=\"325\" height=\"246\" \/> Figure 3. The domain of this function is [latex]x\\ne 10.[\/latex][\/caption]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571597473\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170571597476\" class=\"exercise\">\r\n<div id=\"fs-id1170571597479\" class=\"textbox\">\r\n<p id=\"fs-id1170571597481\">Find the antiderivative of [latex]\\frac{1}{x+2}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572480516\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572480516\"]\r\n<p id=\"fs-id1170572480516\">[latex]\\text{ln}|x+2|+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170573721513\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170572480506\">Follow the pattern from <a class=\"autogenerated-content\" href=\"#fs-id1170571653392\">(Figure)<\/a> to solve the problem.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572480542\" class=\"textbox examples\">\r\n<h3>Finding an Antiderivative of a Rational Function<\/h3>\r\n<div id=\"fs-id1170572480545\" class=\"exercise\">\r\n<div id=\"fs-id1170572480547\" class=\"textbox\">\r\n<p id=\"fs-id1170572480552\">Find the antiderivative of [latex]\\frac{2{x}^{3}+3x}{{x}^{4}+3{x}^{2}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\nThis can be rewritten as [latex]\\int (2{x}^{3}+3x){({x}^{4}+3{x}^{2})}^{-1}dx.[\/latex] Use substitution. Let [latex]u={x}^{4}+3{x}^{2},[\/latex] then [latex]du=4{x}^{3}+6x.[\/latex] Alter <em>du<\/em> by factoring out the 2. Thus,\r\n<div id=\"fs-id1170571807881\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\hfill du&amp; =\\hfill &amp; (4{x}^{3}+6x)dx\\hfill \\\\ &amp; =\\hfill &amp; 2(2{x}^{3}+3x)dx\\hfill \\\\ \\hfill \\frac{1}{2}du&amp; =\\hfill &amp; (2{x}^{3}+3x)dx.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572373464\">Rewrite the integrand in [latex]u[\/latex]:<\/p>\r\n\r\n<div id=\"fs-id1170572331833\" class=\"equation unnumbered\">[latex]\\int (2{x}^{3}+3x){({x}^{4}+3{x}^{2})}^{-1}dx=\\frac{1}{2}\\int {u}^{-1}du.[\/latex]<\/div>\r\n<p id=\"fs-id1170571733965\">Then we have<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ll}\\frac{1}{2}\\int {u}^{-1}du\\hfill &amp; =\\frac{1}{2}\\text{ln}|u|+C\\hfill \\\\ \\\\ &amp; =\\frac{1}{2}\\text{ln}|{x}^{4}+3{x}^{2}|+C.\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571649922\" class=\"textbox examples\">\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n<h3>Finding an Antiderivative of a Logarithmic Function<\/h3>\r\n<p id=\"fs-id1170571649932\">Find the antiderivative of the log function [latex]{\\text{log}}_{2}x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571649954\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571649954\"]\r\n<p id=\"fs-id1170571649954\">Follow the format in the formula listed in the rule on integration formulas involving logarithmic functions. Based on this format, we have<\/p>\r\n\r\n<div id=\"fs-id1170571649959\" class=\"equation unnumbered\">[latex]\\int {\\text{log}}_{2}xdx=\\frac{x}{\\text{ln}2}(\\text{ln}x-1)+C.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571660181\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170571660184\" class=\"exercise\">\r\n<div id=\"fs-id1170571660186\" class=\"textbox\">\r\n<p id=\"fs-id1170571660188\">Find the antiderivative of [latex]{\\text{log}}_{3}x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571660214\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571660214\"]\r\n<p id=\"fs-id1170571660214\">[latex]\\frac{x}{\\text{ln}3}(\\text{ln}x-1)+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170573418990\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170571660204\">Follow <a class=\"autogenerated-content\" href=\"#fs-id1170571649922\">(Figure)<\/a> and refer to the rule on integration formulas involving logarithmic functions.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572274900\"><a class=\"autogenerated-content\" href=\"#fs-id1170572274908\">(Figure)<\/a> is a definite integral of a trigonometric function. With trigonometric functions, we often have to apply a trigonometric property or an identity before we can move forward. Finding the right form of the integrand is usually the key to a smooth integration.<\/p>\r\n\r\n<div id=\"fs-id1170572274908\" class=\"textbox examples\">\r\n<h3>Evaluating a Definite Integral<\/h3>\r\n<div id=\"fs-id1170572274910\" class=\"exercise\">\r\n<div id=\"fs-id1170572274912\" class=\"textbox\">\r\n<p id=\"fs-id1170572274917\">Find the definite integral of [latex]{\\int }_{0}^{\\pi \\text{\/}2}\\frac{ \\sin x}{1+ \\cos x}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571636148\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571636148\"]\r\n<p id=\"fs-id1170571636148\">We need substitution to evaluate this problem. Let [latex]u=1+ \\cos x,,[\/latex] so [latex]du=\\text{\u2212} \\sin xdx.[\/latex] Rewrite the integral in terms of [latex]u[\/latex], changing the limits of integration as well. Thus,<\/p>\r\n\r\n<div id=\"fs-id1170571636203\" class=\"equation unnumbered\">[latex]\\begin{array}{c}u=1+ \\cos (0)=2\\hfill \\\\ u=1+ \\cos (\\frac{\\pi }{2})=1.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572510108\">Then<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\begin{array}{cc}{\\int }_{0}^{\\pi \\text{\/}2}\\frac{ \\sin x}{1+ \\cos x}\\hfill &amp; =\\text{\u2212}{\\int }_{2}^{1}{u}^{-1}du\\hfill \\\\ \\\\ \\\\ &amp; ={\\int }_{1}^{2}{u}^{-1}du\\hfill \\\\ &amp; ={\\text{ln}|u||}_{1}^{2}\\hfill \\\\ &amp; =\\left[\\text{ln}2-\\text{ln}1\\right]\\hfill \\\\ &amp; =\\text{ln}2.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571712566\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1170571712573\">\r\n \t<li>Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay.<\/li>\r\n \t<li>Substitution is often used to evaluate integrals involving exponential functions or logarithms.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170571712590\" class=\"key-equations\">\r\n<h1>Key Equations<\/h1>\r\n<ul>\r\n \t<li><strong>Integrals of Exponential Functions<\/strong>\r\n[latex]\\int {e}^{x}dx={e}^{x}+C[\/latex]\r\n[latex]\\int {a}^{x}dx=\\frac{{a}^{x}}{\\text{ln}a}+C[\/latex]<\/li>\r\n \t<li><strong>Integration Formulas Involving Logarithmic Functions<\/strong>\r\n[latex]\\int {x}^{-1}dx=\\text{ln}|x|+C[\/latex]\r\n[latex]\\int \\text{ln}xdx=x\\text{ln}x-x+C=x(\\text{ln}x-1)+C[\/latex]\r\n[latex]\\int {\\text{log}}_{a}xdx=\\frac{x}{\\text{ln}a}(\\text{ln}x-1)+C[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170571699478\" class=\"textbox exercises\">\r\n<p id=\"fs-id1170571699482\">In the following exercises, compute each indefinite integral.<\/p>\r\n\r\n<div id=\"fs-id1170571699485\" class=\"exercise\">\r\n<div id=\"fs-id1170571699487\" class=\"textbox\">\r\n<p id=\"fs-id1170571699489\">[latex]\\int {e}^{2x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571678838\" class=\"exercise\">\r\n<div id=\"fs-id1170571678840\" class=\"textbox\">\r\n<p id=\"fs-id1170571678842\">[latex]\\int {e}^{-3x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571678870\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571678870\"]\r\n<p id=\"fs-id1170571678870\">[latex]\\frac{-1}{3}{e}^{-3x}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1170572601243\" class=\"textbox\">\r\n<p id=\"fs-id1170572601245\">[latex]\\int {2}^{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572601296\" class=\"exercise\">\r\n<div id=\"fs-id1170572601298\" class=\"textbox\">\r\n<p id=\"fs-id1170572601300\">[latex]\\int {3}^{\\text{\u2212}x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571679785\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571679785\"]\r\n<p id=\"fs-id1170571679785\">[latex]-\\frac{{3}^{\\text{\u2212}x}}{\\text{ln}3}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571679817\" class=\"exercise\">\r\n<div id=\"fs-id1170571679819\" class=\"textbox\">\r\n<p id=\"fs-id1170571679821\">[latex]\\int \\frac{1}{2x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572611839\" class=\"exercise\">\r\n<div id=\"fs-id1170572611841\" class=\"textbox\">\r\n<p id=\"fs-id1170572611843\">[latex]\\int \\frac{2}{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572611870\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572611870\"]\r\n<p id=\"fs-id1170572611870\">[latex]\\text{ln}({x}^{2})+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572611894\" class=\"exercise\">\r\n<div id=\"fs-id1170572611896\" class=\"textbox\">\r\n\r\n[latex]\\int \\frac{1}{{x}^{2}}dx[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572333073\" class=\"exercise\">\r\n<div id=\"fs-id1170572333075\" class=\"textbox\">\r\n<p id=\"fs-id1170572333077\">[latex]\\int \\frac{1}{\\sqrt{x}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572333105\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572333105\"]\r\n<p id=\"fs-id1170572333105\">[latex]2\\sqrt{x}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572333121\">In the following exercises, find each indefinite integral by using appropriate substitutions.<\/p>\r\n\r\n<div id=\"fs-id1170572333125\" class=\"exercise\">\r\n<div id=\"fs-id1170572333127\" class=\"textbox\">\r\n<p id=\"fs-id1170572333129\">[latex]\\int \\frac{\\text{ln}x}{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571638228\" class=\"exercise\">\r\n<div id=\"fs-id1170571638230\" class=\"textbox\">\r\n<p id=\"fs-id1170571638233\">[latex]\\int \\frac{dx}{x{(\\text{ln}x)}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571810853\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571810853\"]\r\n<p id=\"fs-id1170571810853\">[latex]-\\frac{1}{\\text{ln}x}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571810876\" class=\"exercise\">\r\n<div id=\"fs-id1170571810878\" class=\"textbox\">\r\n<p id=\"fs-id1170571810880\">[latex]\\int \\frac{dx}{x\\text{ln}x}(x&gt;1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571697166\" class=\"exercise\">\r\n<div id=\"fs-id1170571697168\" class=\"textbox\">\r\n<p id=\"fs-id1170571697170\">[latex]\\int \\frac{dx}{x\\text{ln}x\\text{ln}(\\text{ln}x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571697227\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571697227\"]\r\n<p id=\"fs-id1170571697227\">[latex]\\text{ln}(\\text{ln}(\\text{ln}x))+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571539147\" class=\"exercise\">\r\n<div id=\"fs-id1170571539149\" class=\"textbox\">\r\n<p id=\"fs-id1170571539151\">[latex]\\int \\tan \\theta d\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571539202\" class=\"exercise\">\r\n<div id=\"fs-id1170571539204\" class=\"textbox\">\r\n<p id=\"fs-id1170571539206\">[latex]\\int \\frac{ \\cos x-x \\sin x}{x \\cos x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170571543234\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571543234\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571543234\"][latex]\\text{ln}(x \\cos x)+C[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571543268\" class=\"exercise\">\r\n<div id=\"fs-id1170571543270\" class=\"textbox\">\r\n<p id=\"fs-id1170571543272\">[latex]\\int \\frac{\\text{ln}( \\sin x)}{ \\tan x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572456410\" class=\"exercise\">\r\n<div id=\"fs-id1170572456412\" class=\"textbox\">\r\n<p id=\"fs-id1170572456414\">[latex]\\int \\text{ln}( \\cos x) \\tan xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\n[latex]-\\frac{1}{2}{(\\text{ln}( \\cos (x)))}^{2}+C[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572396562\" class=\"exercise\">\r\n<div id=\"fs-id1170572396564\" class=\"textbox\">\r\n<p id=\"fs-id1170572396567\">[latex]\\int x{e}^{\\text{\u2212}{x}^{2}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1170571580982\">[latex]\\int {x}^{2}{e}^{\\text{\u2212}{x}^{3}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572218578\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572218578\"]\r\n<p id=\"fs-id1170572218578\">[latex]\\frac{\\text{\u2212}{e}^{\\text{\u2212}{x}^{3}}}{3}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218606\" class=\"exercise\">\r\n<div id=\"fs-id1170572218608\" class=\"textbox\">\r\n<p id=\"fs-id1170572218611\">[latex]\\int {e}^{ \\sin x} \\cos xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218671\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\n[latex]\\int {e}^{ \\tan x}{ \\sec }^{2}xdx[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572386146\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572386146\"]\r\n<p id=\"fs-id1170572386146\">[latex]{e}^{ \\tan x}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572386168\" class=\"exercise\">\r\n<div id=\"fs-id1170572386170\" class=\"textbox\">\r\n<p id=\"fs-id1170572386172\">[latex]\\int {e}^{\\text{ln}x}\\frac{dx}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\n[latex]\\int \\frac{{e}^{\\text{ln}(1-t)}}{1-t}dt[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572643259\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572643259\"]\r\n<p id=\"fs-id1170572643259\">[latex]t+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nIn the following exercises, verify by differentiation that [latex]\\int \\text{ln}xdx=x(\\text{ln}x-1)+C,[\/latex] then use appropriate changes of variables to compute the integral.\r\n<div id=\"fs-id1170572373699\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\n[latex]\\int \\text{ln}xdx[\/latex][latex](Hint\\text{:}\\int \\text{ln}xdx=\\frac{1}{2}\\int x\\text{ln}({x}^{2})dx)[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571710720\" class=\"exercise\">\r\n<div id=\"fs-id1170571710722\" class=\"textbox\">\r\n<p id=\"fs-id1170571710724\">[latex]\\int {x}^{2}{\\text{ln}}^{2}xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572399000\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572399000\"]\r\n<p id=\"fs-id1170572399000\">[latex]\\frac{1}{9}{x}^{3}(\\text{ln}({x}^{3})-1)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572399045\" class=\"exercise\">\r\n<div id=\"fs-id1170572399048\" class=\"textbox\">\r\n<p id=\"fs-id1170572399050\">[latex]\\int \\frac{\\text{ln}x}{{x}^{2}}dx[\/latex][latex](Hint\\text{:}\\text{Set}u=\\frac{1}{x}\\text{.})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572415189\" class=\"exercise\">\r\n<div id=\"fs-id1170572415191\" class=\"textbox\">\r\n<p id=\"fs-id1170572415193\">[latex]\\int \\frac{\\text{ln}x}{\\sqrt{x}}dx[\/latex][latex](Hint\\text{:}\\text{Set}u=\\sqrt{x}\\text{.})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572168712\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572168712\"]\r\n<p id=\"fs-id1170572168712\">[latex]2\\sqrt{x}(\\text{ln}x-2)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1170572168748\" class=\"textbox\">\r\n<p id=\"fs-id1170572168750\">Write an integral to express the area under the graph of [latex]y=\\frac{1}{t}[\/latex] from [latex]t=1[\/latex] to <em>e<sup>x<\/sup><\/em> and evaluate the integral.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572629242\" class=\"exercise\">\r\n<div id=\"fs-id1170572629244\" class=\"textbox\">\r\n<p id=\"fs-id1170572629246\">Write an integral to express the area under the graph of [latex]y={e}^{t}[\/latex] between [latex]t=0[\/latex] and [latex]t=\\text{ln}x,[\/latex] and evaluate the integral.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571689752\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571689752\"]\r\n<p id=\"fs-id1170571689752\">[latex]{\\int }_{0}^{\\text{ln}x}{e}^{t}dt={e}^{t}{|}_{0}^{\\text{ln}x}={e}^{\\text{ln}x}-{e}^{0}=x-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572569960\">In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.<\/p>\r\n\r\n<div id=\"fs-id1170572569965\" class=\"exercise\">\r\n<div id=\"fs-id1170572569967\" class=\"textbox\">\r\n<p id=\"fs-id1170572569969\">[latex]\\int \\tan (2x)dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571571932\" class=\"exercise\">\r\n<div id=\"fs-id1170571571934\" class=\"textbox\">\r\n<p id=\"fs-id1170571571936\">[latex]\\int \\frac{ \\sin (3x)- \\cos (3x)}{ \\sin (3x)+ \\cos (3x)}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571572016\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571572016\"]\r\n<p id=\"fs-id1170571572016\">[latex]-\\frac{1}{3}\\text{ln}( \\sin (3x)+ \\cos (3x))[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572379018\" class=\"exercise\">\r\n<div id=\"fs-id1170572379020\" class=\"textbox\">\r\n<p id=\"fs-id1170572379022\">[latex]\\int \\frac{x \\sin ({x}^{2})}{ \\cos ({x}^{2})}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571777887\" class=\"exercise\">\r\n<div id=\"fs-id1170571777889\" class=\"textbox\">\r\n<p id=\"fs-id1170571777891\">[latex]\\int x \\csc ({x}^{2})dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571777931\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571777931\"]\r\n<p id=\"fs-id1170571777931\">[latex]-\\frac{1}{2}\\text{ln}| \\csc ({x}^{2})+ \\cot ({x}^{2})|+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572310019\" class=\"exercise\">\r\n<div id=\"fs-id1170572310021\" class=\"textbox\">\r\n<p id=\"fs-id1170572310023\">[latex]\\int \\text{ln}( \\cos x) \\tan xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571613744\" class=\"exercise\">\r\n<div id=\"fs-id1170571613746\" class=\"textbox\">\r\n<p id=\"fs-id1170571613748\">[latex]\\int \\text{ln}( \\csc x) \\cot xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571613792\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571613792\"]\r\n<p id=\"fs-id1170571613792\">[latex]-\\frac{1}{2}{(\\text{ln}( \\csc x))}^{2}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572451363\" class=\"exercise\">\r\n<div id=\"fs-id1170572451366\" class=\"textbox\">\r\n<p id=\"fs-id1170572451368\">[latex]\\int \\frac{{e}^{x}-{e}^{\\text{\u2212}x}}{{e}^{x}+{e}^{\\text{\u2212}x}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572346997\">In the following exercises, evaluate the definite integral.<\/p>\r\n\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1170572347004\">[latex]{\\int }_{1}^{2}\\frac{1+2x+{x}^{2}}{3x+3{x}^{2}+{x}^{3}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572347069\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572347069\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572347069\"][latex]\\frac{1}{3}\\text{ln}(\\frac{26}{7})[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572329945\" class=\"exercise\">\r\n<div id=\"fs-id1170572329947\" class=\"textbox\">\r\n<p id=\"fs-id1170572329949\">[latex]{\\int }_{0}^{\\pi \\text{\/}4} \\tan xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572330008\" class=\"exercise\">\r\n<div id=\"fs-id1170572330010\" class=\"textbox\">\r\n<p id=\"fs-id1170572330012\">[latex]{\\int }_{0}^{\\pi \\text{\/}3}\\frac{ \\sin x- \\cos x}{ \\sin x+ \\cos x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571613655\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571613655\"]\r\n<p id=\"fs-id1170571613655\">[latex]\\text{ln}(\\sqrt{3}-1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571613677\" class=\"exercise\">\r\n<div id=\"fs-id1170571613679\" class=\"textbox\">\r\n<p id=\"fs-id1170571613681\">[latex]{\\int }_{\\pi \\text{\/}6}^{\\pi \\text{\/}2} \\csc xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\n[latex]{\\int }_{\\pi \\text{\/}4}^{\\pi \\text{\/}3} \\cot xdx[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572412258\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572412258\"]\r\n<p id=\"fs-id1170572412258\">[latex]\\frac{1}{2}\\text{ln}\\frac{3}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572412280\">In the following exercises, integrate using the indicated substitution.<\/p>\r\n\r\n<div id=\"fs-id1170572412283\" class=\"exercise\">\r\n<div id=\"fs-id1170572412285\" class=\"textbox\">\r\n<p id=\"fs-id1170572412287\">[latex]\\int \\frac{x}{x-100}dx;u=x-100[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572627084\" class=\"exercise\">\r\n<div id=\"fs-id1170572627086\" class=\"textbox\">\r\n<p id=\"fs-id1170572627088\">[latex]\\int \\frac{y-1}{y+1}dy;u=y+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572627138\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572627138\"]\r\n<p id=\"fs-id1170572627138\">[latex]y-2\\text{ln}|y+1|+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572627173\" class=\"exercise\">\r\n<div id=\"fs-id1170572627176\" class=\"textbox\">\r\n<p id=\"fs-id1170572572238\">[latex]\\int \\frac{1-{x}^{2}}{3x-{x}^{3}}dx;u=3x-{x}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572572338\" class=\"exercise\">\r\n<div id=\"fs-id1170572572340\" class=\"textbox\">\r\n\r\n[latex]\\int \\frac{ \\sin x+ \\cos x}{ \\sin x- \\cos x}dx;u= \\sin x- \\cos x[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571712850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571712850\"]\r\n<p id=\"fs-id1170571712850\">[latex]\\text{ln}| \\sin x- \\cos x|+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572129747\" class=\"exercise\">\r\n<div id=\"fs-id1170572129749\" class=\"textbox\">\r\n<p id=\"fs-id1170572129752\">[latex]\\int {e}^{2x}\\sqrt{1-{e}^{2x}}dx;u={e}^{2x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572379189\" class=\"exercise\">\r\n<div id=\"fs-id1170572379191\" class=\"textbox\">\r\n<p id=\"fs-id1170572379193\">[latex]\\int \\text{ln}(x)\\frac{\\sqrt{1-{(\\text{ln}x)}^{2}}}{x}dx;u=\\text{ln}x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572379266\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572379266\"]\r\n<p id=\"fs-id1170572379266\">[latex]-\\frac{1}{3}{(1-(\\text{ln}{x}^{2}))}^{3\\text{\/}2}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572330224\">In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate <em>R<\/em><sub>50<\/sub> and solve for the exact area.<\/p>\r\n\r\n<div id=\"fs-id1170572330236\" class=\"exercise\">\r\n<div id=\"fs-id1170572330238\" class=\"textbox\">\r\n<p id=\"fs-id1170572330240\"><strong>[T]<\/strong>[latex]y={e}^{x}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572129062\" class=\"exercise\">\r\n<div id=\"fs-id1170572129064\" class=\"textbox\">\r\n<p id=\"fs-id1170572129066\"><strong>[T]<\/strong>[latex]y={e}^{\\text{\u2212}x}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572129108\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572129108\"]\r\n<p id=\"fs-id1170572129108\">Exact solution: [latex]\\frac{e-1}{e},{R}_{50}=0.6258.[\/latex] Since [latex]f[\/latex] is decreasing, the right endpoint estimate underestimates the area.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572129147\" class=\"exercise\">\r\n<div id=\"fs-id1170572129149\" class=\"textbox\">\r\n<p id=\"fs-id1170572129152\"><strong>[T]<\/strong>[latex]y=\\text{ln}(x)[\/latex] over [latex]\\left[1,2\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572223556\" class=\"exercise\">\r\n<div id=\"fs-id1170572223558\" class=\"textbox\">\r\n<p id=\"fs-id1170572223560\"><strong>[T]<\/strong>[latex]y=\\frac{x+1}{{x}^{2}+2x+6}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571568951\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571568951\"]\r\n<p id=\"fs-id1170571568951\">Exact solution: [latex]\\frac{2\\text{ln}(3)-\\text{ln}(6)}{2},{R}_{50}=0.2033.[\/latex] Since [latex]f[\/latex] is increasing, the right endpoint estimate overestimates the area.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571569011\" class=\"exercise\">\r\n<div id=\"fs-id1170571569013\" class=\"textbox\">\r\n<p id=\"fs-id1170571569015\"><strong>[T]<\/strong>[latex]y={2}^{x}[\/latex] over [latex]\\left[-1,0\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572309777\" class=\"exercise\">\r\n<div id=\"fs-id1170572309779\" class=\"textbox\">\r\n<p id=\"fs-id1170572309781\"><strong>[T]<\/strong>[latex]y=\\text{\u2212}{2}^{\\text{\u2212}x}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571628910\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571628910\"]\r\n<p id=\"fs-id1170571628910\">Exact solution: [latex]-\\frac{1}{\\text{ln}(4)},{R}_{50}=-0.7164.[\/latex] Since [latex]f[\/latex] is increasing, the right endpoint estimate overestimates the area (the actual area is a larger negative number).<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170571628956\">In the following exercises, [latex]f(x)\\ge 0[\/latex] for [latex]a\\le x\\le b.[\/latex] Find the area under the graph of [latex]f(x)[\/latex] between the given values [latex]a[\/latex] and [latex]b[\/latex] by integrating.<\/p>\r\n\r\n<div id=\"fs-id1170572582554\" class=\"exercise\">\r\n<div id=\"fs-id1170572582556\" class=\"textbox\">\r\n<p id=\"fs-id1170572582558\">[latex]f(x)=\\frac{{\\text{log}}_{10}(x)}{x};a=10,b=100[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572582636\" class=\"exercise\">\r\n<div id=\"fs-id1170572582638\" class=\"textbox\">\r\n<p id=\"fs-id1170572582641\">[latex]f(x)=\\frac{{\\text{log}}_{2}(x)}{x};a=32,b=64[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572351521\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572351521\"]\r\n<p id=\"fs-id1170572351521\">[latex]\\frac{11}{2}\\text{ln}2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572351541\" class=\"exercise\">\r\n<div id=\"fs-id1170572351543\" class=\"textbox\">\r\n<p id=\"fs-id1170572351545\">[latex]f(x)={2}^{\\text{\u2212}x};a=1,b=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571638055\" class=\"exercise\">\r\n<div id=\"fs-id1170571638057\" class=\"textbox\">\r\n<p id=\"fs-id1170571638060\">[latex]f(x)={2}^{\\text{\u2212}x};a=3,b=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571638104\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571638104\"]\r\n<p id=\"fs-id1170571638104\">[latex]\\frac{1}{\\text{ln}(65,536)}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571638130\" class=\"exercise\">\r\n<div id=\"fs-id1170571638132\" class=\"textbox\">\r\n<p id=\"fs-id1170571638134\">Find the area under the graph of the function [latex]f(x)=x{e}^{\\text{\u2212}{x}^{2}}[\/latex] between [latex]x=0[\/latex] and [latex]x=5.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571610335\" class=\"exercise\">\r\n<div id=\"fs-id1170571610337\" class=\"textbox\">\r\n<p id=\"fs-id1170571610339\">Compute the integral of [latex]f(x)=x{e}^{\\text{\u2212}{x}^{2}}[\/latex] and find the smallest value of <em>N<\/em> such that the area under the graph [latex]f(x)=x{e}^{\\text{\u2212}{x}^{2}}[\/latex] between [latex]x=N[\/latex] and [latex]x=N+10[\/latex] is, at most, 0.01.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572504510\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572504510\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572504510\"][latex]{\\int }_{N}^{N+1}x{e}^{\\text{\u2212}{x}^{2}}dx=\\frac{1}{2}({e}^{\\text{\u2212}{N}^{2}}-{e}^{\\text{\u2212}{(N+1)}^{2}}).[\/latex] The quantity is less than 0.01 when [latex]N=2.[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572368492\" class=\"exercise\">\r\n<div id=\"fs-id1170572368494\" class=\"textbox\">\r\n<p id=\"fs-id1170572368496\">Find the limit, as <em>N<\/em> tends to infinity, of the area under the graph of [latex]f(x)=x{e}^{\\text{\u2212}{x}^{2}}[\/latex] between [latex]x=0[\/latex] and [latex]x=5.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572448465\" class=\"exercise\">\r\n<div id=\"fs-id1170572448467\" class=\"textbox\">\r\n<p id=\"fs-id1170572448469\">Show that [latex]{\\int }_{a}^{b}\\frac{dt}{t}={\\int }_{1\\text{\/}b}^{1\\text{\/}a}\\frac{dt}{t}[\/latex] when [latex]0&lt;a\\le b.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572306386\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572306386\"]\r\n<p id=\"fs-id1170572306386\">[latex]{\\int }_{a}^{b}\\frac{dx}{x}=\\text{ln}(b)-\\text{ln}(a)=\\text{ln}(\\frac{1}{a})-\\text{ln}(\\frac{1}{b})={\\int }_{1\\text{\/}b}^{1\\text{\/}a}\\frac{dx}{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572503285\" class=\"exercise\">\r\n<div id=\"fs-id1170572503288\" class=\"textbox\">\r\n<p id=\"fs-id1170572503290\">Suppose that [latex]f(x)&gt;0[\/latex] for all [latex]x[\/latex] and that [latex]f[\/latex] and [latex]g[\/latex] are differentiable. Use the identity [latex]{f}^{g}={e}^{g\\text{ln}f}[\/latex] and the chain rule to find the derivative of [latex]{f}^{g}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1170572558121\" class=\"textbox\">\r\n<p id=\"fs-id1170572558123\">Use the previous exercise to find the antiderivative of [latex]h(x)={x}^{x}(1+\\text{ln}x)[\/latex] and evaluate [latex]{\\int }_{2}^{3}{x}^{x}(1+\\text{ln}x)dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571814004\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571814004\"]\r\n<p id=\"fs-id1170571814004\">23<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571814009\" class=\"exercise\">\r\n<div id=\"fs-id1170571814011\" class=\"textbox\">\r\n<p id=\"fs-id1170571814013\">Show that if [latex]c&gt;0,[\/latex] then the integral of [latex]1\\text{\/}x[\/latex] from <em>ac<\/em> to <em>bc<\/em> [latex](0&lt;a&lt;b)[\/latex] is the same as the integral of [latex]1\\text{\/}x[\/latex] from [latex]a[\/latex] to [latex]b[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572307624\">The following exercises are intended to derive the fundamental properties of the natural log starting from the <em>Definition\/em&gt; [latex]\\text{ln}(x)={\\int }_{1}^{x}\\frac{dt}{t},[\/latex] using properties of the definite integral and making no further assumptions.<\/em><\/p>\r\n\r\n<div id=\"fs-id1170572307674\" class=\"exercise\">\r\n<div id=\"fs-id1170572307676\" class=\"textbox\">\r\n<p id=\"fs-id1170572307678\">Use the identity [latex]\\text{ln}(x)={\\int }_{1}^{x}\\frac{dt}{t}[\/latex] to derive the identity [latex]\\text{ln}(\\frac{1}{x})=\\text{\u2212}\\text{ln}x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572296602\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572296602\"]\r\n<p id=\"fs-id1170572296602\">We may assume that [latex]x&gt;1,\\text{so}\\frac{1}{x}&lt;1.[\/latex] Then, [latex]{\\int }_{1}^{1\\text{\/}x}\\frac{dt}{t}.[\/latex] Now make the substitution [latex]u=\\frac{1}{t},[\/latex] so [latex]du=-\\frac{dt}{{t}^{2}}[\/latex] and [latex]\\frac{du}{u}=-\\frac{dt}{t},[\/latex] and change endpoints: [latex]{\\int }_{1}^{1\\text{\/}x}\\frac{dt}{t}=\\text{\u2212}{\\int }_{1}^{x}\\frac{du}{u}=\\text{\u2212}\\text{ln}x.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\nUse a change of variable in the integral [latex]{\\int }_{1}^{xy}\\frac{1}{t}dt[\/latex] to show that [latex]\\text{ln}xy=\\text{ln}x+\\text{ln}y\\text{ for }x,y&gt;0.[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571629695\" class=\"exercise\">\r\n<div id=\"fs-id1170571629697\" class=\"textbox\">\r\n<p id=\"fs-id1170571629699\">Use the identity [latex]\\text{ln}x={\\int }_{1}^{x}\\frac{dt}{x}[\/latex] to show that [latex]\\text{ln}(x)[\/latex] is an increasing function of [latex]x[\/latex] on [latex]\\left[0,\\infty ),[\/latex] and use the previous exercises to show that the range of [latex]\\text{ln}(x)[\/latex] is [latex](\\text{\u2212}\\infty ,\\infty ).[\/latex] Without any further assumptions, conclude that [latex]\\text{ln}(x)[\/latex] has an inverse function defined on [latex](\\text{\u2212}\\infty ,\\infty ).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571624156\" class=\"exercise\">\r\n<div id=\"fs-id1170571624158\" class=\"textbox\">\r\n<p id=\"fs-id1170571624160\">Pretend, for the moment, that we do not know that [latex]{e}^{x}[\/latex] is the inverse function of [latex]\\text{ln}(x),[\/latex] but keep in mind that [latex]\\text{ln}(x)[\/latex] has an inverse function defined on [latex](\\text{\u2212}\\infty ,\\infty ).[\/latex] Call it <em>E<\/em>. Use the identity [latex]\\text{ln}xy=\\text{ln}x+\\text{ln}y[\/latex] to deduce that [latex]E(a+b)=E(a)E(b)[\/latex] for any real numbers [latex]a[\/latex], [latex]b[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572554258\" class=\"exercise\">\r\n<div id=\"fs-id1170572554260\" class=\"textbox\">\r\n<p id=\"fs-id1170572554263\">Pretend, for the moment, that we do not know that [latex]{e}^{x}[\/latex] is the inverse function of [latex]\\text{ln}x,[\/latex] but keep in mind that [latex]\\text{ln}x[\/latex] has an inverse function defined on [latex](\\text{\u2212}\\infty ,\\infty ).[\/latex] Call it <em>E<\/em>. Show that [latex]E\\text{'}(t)=E(t).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572309577\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572309577\"]\r\n<p id=\"fs-id1170572309577\">[latex]x=E(\\text{ln}(x)).[\/latex] Then, [latex]1=\\frac{E\\text{'}(\\text{ln}x)}{x}\\text{or}x=E\\text{'}(\\text{ln}x).[\/latex] Since any number [latex]t[\/latex] can be written [latex]t=\\text{ln}x[\/latex] for some [latex]x[\/latex], and for such [latex]t[\/latex] we have [latex]x=E(t),[\/latex] it follows that for any [latex]t,E\\text{'}(t)=E(t).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1170571612025\" class=\"textbox\">\r\n<p id=\"fs-id1170571612028\">The sine integral, defined as [latex]S(x)={\\int }_{0}^{x}\\frac{ \\sin t}{t}dt[\/latex] is an important quantity in engineering. Although it does not have a simple closed formula, it is possible to estimate its behavior for large [latex]x[\/latex]. Show that for [latex]k\\ge 1,|S(2\\pi k)-S(2\\pi (k+1))|\\le \\frac{1}{k(2k+1)\\pi }.[\/latex] [latex](Hint\\text{:} \\sin (t+\\pi )=\\text{\u2212} \\sin t)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572373482\" class=\"exercise\">\r\n<div id=\"fs-id1170572373484\" class=\"textbox\">\r\n<p id=\"fs-id1170572373486\"><strong>[T]<\/strong> The normal distribution in probability is given by [latex]p(x)=\\frac{1}{\\sigma \\sqrt{2\\pi }}{e}^{\\text{\u2212}{(x-\\mu )}^{2}\\text{\/}2{\\sigma }^{2}},[\/latex] where <em>\u03c3<\/em> is the standard deviation and <em>\u03bc<\/em> is the average. The <em>standard normal distribution<\/em> in probability, [latex]{p}_{s},[\/latex] corresponds to [latex]\\mu =0\\text{ and }\\sigma =1.[\/latex] Compute the left endpoint estimates [latex]{R}_{10}\\text{ and }{R}_{100}[\/latex] of [latex]{\\int }_{-1}^{1}\\frac{1}{\\sqrt{2\\pi }}{e}^{\\text{\u2212}{x}^{2\\text{\/}2}}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571652256\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571652256\"]\r\n<p id=\"fs-id1170571652256\">[latex]{R}_{10}=0.6811,{R}_{100}=0.6827[\/latex]<\/p>\r\n<span id=\"fs-id1170571652285\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204259\/CNX_Calc_Figure_05_06_202.jpg\" alt=\"A graph of the function f(x) = .5 * ( sqrt(2)*e^(-.5x^2)) \/ sqrt(pi). It is a downward opening curve that is symmetric across the y axis, crossing at about (0, .4). It approaches 0 as x goes to positive and negative infinity. Between 1 and -1, ten rectangles are drawn for a right endpoint estimate of the area under the curve.\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572380098\" class=\"exercise\">\r\n<div id=\"fs-id1170572380101\" class=\"textbox\">\r\n<p id=\"fs-id1170572380103\"><strong>[T]<\/strong> Compute the right endpoint estimates [latex]{R}_{50}\\text{ and }{R}_{100}[\/latex] of [latex]{\\int }_{-3}^{5}\\frac{1}{2\\sqrt{2\\pi }}{e}^{\\text{\u2212}{(x-1)}^{2}\\text{\/}8}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Integrate functions involving exponential functions.<\/li>\n<li>Integrate functions involving logarithmic functions.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1170572130673\">Exponential and logarithmic functions are used to model population growth, cell growth, and financial growth, as well as depreciation, radioactive decay, and resource consumption, to name only a few applications. In this section, we explore integration involving exponential and logarithmic functions.<\/p>\n<div id=\"fs-id1170572137421\" class=\"bc-section section\">\n<h1>Integrals of Exponential Functions<\/h1>\n<p id=\"fs-id1170571602573\">The exponential function is perhaps the most efficient function in terms of the operations of calculus. The exponential function, [latex]y={e}^{x},[\/latex] is its own derivative and its own integral.<\/p>\n<div id=\"fs-id1170572370390\" class=\"textbox key-takeaways\">\n<h3>Rule: Integrals of Exponential Functions<\/h3>\n<p id=\"fs-id1170572375243\">Exponential functions can be integrated using the following formulas.<\/p>\n<div id=\"fs-id1170572374424\" class=\"equation\">[latex]\\begin{array}{ccc}\\int {e}^{x}dx\\hfill & =\\hfill & {e}^{x}+C\\hfill \\\\ \\int {a}^{x}dx\\hfill & =\\hfill & \\frac{{a}^{x}}{\\text{ln}a}+C\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id11705722115180\" class=\"textbox examples\">\n<div id=\"fs-id1170572606041\" class=\"exercise\">\n<div id=\"fs-id1170572222389\" class=\"textbox\">\n<h3>Finding an Antiderivative of an Exponential Function<\/h3>\n<p id=\"fs-id1170572150624\">Find the antiderivative of the exponential function [latex]e[\/latex]<sup>\u2212[latex]x[\/latex]<\/sup>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571601500\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571601500\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571601500\">Use substitution, setting [latex]u=\\text{\u2212}x,[\/latex] and then [latex]du=-1dx.[\/latex] Multiply the <em>du<\/em> equation by \u22121, so you now have [latex]\\text{\u2212}du=dx.[\/latex] Then,<\/p>\n<div id=\"fs-id1170572454214\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\int {e}^{\\text{\u2212}x}dx\\hfill & =\\text{\u2212}\\int {e}^{u}du\\hfill \\\\ \\\\ & =\\text{\u2212}{e}^{u}+C\\hfill \\\\ & =\\text{\u2212}{e}^{\\text{\u2212}x}+C.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572248011\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170571655362\" class=\"exercise\">\n<div id=\"fs-id1170572206116\" class=\"textbox\">\n<p id=\"fs-id1170572430365\">Find the antiderivative of the function using substitution: [latex]{x}^{2}{e}^{-2{x}^{3}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572269592\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572269592\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572269592\">[latex]\\int {x}^{2}{e}^{-2{x}^{3}}dx=-\\frac{1}{6}{e}^{-2{x}^{3}}+C[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170573205217\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170571622632\">Let [latex]u[\/latex] equal the exponent on [latex]e[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572607911\">A common mistake when dealing with exponential expressions is treating the exponent on [latex]e[\/latex] the same way we treat exponents in polynomial expressions. We cannot use the power rule for the exponent on [latex]e[\/latex]. This can be especially confusing when we have both exponentials and polynomials in the same expression, as in the previous checkpoint. In these cases, we should always double-check to make sure we\u2019re using the right rules for the functions we\u2019re integrating.<\/p>\n<div id=\"fs-id1170572209244\" class=\"textbox examples\">\n<h3>Square Root of an Exponential Function<\/h3>\n<div id=\"fs-id1170571602248\" class=\"exercise\">\n<div id=\"fs-id1170572509298\" class=\"textbox\">\n<p id=\"fs-id1170571773419\">Find the antiderivative of the exponential function [latex]{e}^{x}\\sqrt{1+{e}^{x}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572551693\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572551693\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572551693\">First rewrite the problem using a rational exponent:<\/p>\n<div id=\"fs-id1170572549246\" class=\"equation unnumbered\">[latex]\\int {e}^{x}\\sqrt{1+{e}^{x}}dx=\\int {e}^{x}{(1+{e}^{x})}^{1\\text{\/}2}dx.[\/latex]<\/div>\n<p>Using substitution, choose [latex]u=1+{e}^{x}.u=1+{e}^{x}.[\/latex] Then, [latex]du={e}^{x}dx.[\/latex] We have (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_05_06_001\">(Figure)<\/a>)<\/p>\n<div id=\"fs-id1170572373474\" class=\"equation unnumbered\">[latex]\\int {e}^{x}{(1+{e}^{x})}^{1\\text{\/}2}dx=\\int {u}^{1\\text{\/}2}du.[\/latex]<\/div>\n<p id=\"fs-id1170572307548\">Then<\/p>\n<div id=\"fs-id1170572560605\" class=\"equation unnumbered\">[latex]\\int {u}^{1\\text{\/}2}du=\\frac{{u}^{3\\text{\/}2}}{3\\text{\/}2}+C=\\frac{2}{3}{u}^{3\\text{\/}2}+C=\\frac{2}{3}{(1+{e}^{x})}^{3\\text{\/}2}+C.[\/latex]<\/div>\n<div id=\"CNX_Calc_Figure_05_06_001\" class=\"wp-caption aligncenter\">\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204252\/CNX_Calc_Figure_05_06_001.jpg\" alt=\"A graph of the function f(x) = e^x * sqrt(1 + e^x), which is an increasing concave up curve, over &#091;-3, 1&#093;. It begins close to the x axis in quadrant two, crosses the y axis at (0, sqrt(2)), and continues to increase rapidly.\" width=\"325\" height=\"208\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. The graph shows an exponential function times the square root of an exponential function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572207056\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572451630\" class=\"exercise\">\n<div id=\"fs-id1170571609364\" class=\"textbox\">\n<p id=\"fs-id1170572363371\">Find the antiderivative of [latex]{e}^{x}{(3{e}^{x}-2)}^{2}.[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p id=\"fs-id1170572209127\">[latex]\\int {e}^{x}{(3{e}^{x}-2)}^{2}dx=\\frac{1}{9}{(3{e}^{x}-2)}^{3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170573502097\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p>Let [latex]u=3{e}^{x}-2u=3{e}^{x}-2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1170572240614\" class=\"exercise\">\n<div id=\"fs-id1170572209184\" class=\"textbox\">\n<h3>Using Substitution with an Exponential Function<\/h3>\n<p>Use substitution to evaluate the indefinite integral [latex]\\int 3{x}^{2}{e}^{2{x}^{3}}dx.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572216534\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572216534\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572216534\">Here we choose to let [latex]u[\/latex] equal the expression in the exponent on [latex]e[\/latex]. Let [latex]u=2{x}^{3}[\/latex] and [latex]du=6{x}^{2}dx..[\/latex] Again, <em>du<\/em> is off by a constant multiplier; the original function contains a factor of 3[latex]x[\/latex]<sup>2<\/sup>, not 6[latex]x[\/latex]<sup>2<\/sup>. Multiply both sides of the equation by [latex]\\frac{1}{2}[\/latex] so that the integrand in [latex]u[\/latex] equals the integrand in [latex]x[\/latex]. Thus,<\/p>\n<div id=\"fs-id1170572470447\" class=\"equation unnumbered\">[latex]\\int 3{x}^{2}{e}^{2{x}^{3}}dx=\\frac{1}{2}\\int {e}^{u}du.[\/latex]<\/div>\n<p id=\"fs-id1170572101855\">Integrate the expression in [latex]u[\/latex] and then substitute the original expression in [latex]x[\/latex] back into the [latex]u[\/latex] integral:<\/p>\n<div id=\"fs-id1170572448333\" class=\"equation unnumbered\">[latex]\\frac{1}{2}\\int {e}^{u}du=\\frac{1}{2}{e}^{u}+C=\\frac{1}{2}{e}^{2{x}^{3}}+C.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572551647\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572294922\" class=\"exercise\">\n<div id=\"fs-id1170572095031\" class=\"textbox\">\n<p id=\"fs-id1170572295300\">Evaluate the indefinite integral [latex]\\int 2{x}^{3}{e}^{{x}^{4}}dx.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572242344\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572242344\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572242344\">[latex]\\int 2{x}^{3}{e}^{{x}^{4}}dx=\\frac{1}{2}{e}^{{x}^{4}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170573414530\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170572232621\">Let [latex]u={x}^{4}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170571733816\">As mentioned at the beginning of this section, exponential functions are used in many real-life applications. The number [latex]e[\/latex] is often associated with compounded or accelerating growth, as we have seen in earlier sections about the derivative. Although the derivative represents a rate of change or a growth rate, the integral represents the total change or the total growth. Let\u2019s look at an example in which integration of an exponential function solves a common business application.<\/p>\n<p id=\"fs-id1170571571226\">A <span class=\"no-emphasis\">price\u2013demand function<\/span> tells us the relationship between the quantity of a product demanded and the price of the product. In general, price decreases as quantity demanded increases. The marginal price\u2013demand function is the derivative of the price\u2013demand function and it tells us how fast the price changes at a given level of production. These functions are used in business to determine the price\u2013elasticity of demand, and to help companies determine whether changing production levels would be profitable.<\/p>\n<div id=\"fs-id1170572100406\" class=\"textbox examples\">\n<div id=\"fs-id1170571597871\" class=\"exercise\">\n<div class=\"textbox\">\n<h3>Finding a Price\u2013Demand Equation<\/h3>\n<p id=\"fs-id1170571526652\">Find the price\u2013demand equation for a particular brand of toothpaste at a supermarket chain when the demand is 50 tubes per week at $2.35 per tube, given that the marginal price\u2014demand function, [latex]{p}^{\\prime }(x),[\/latex] for [latex]x[\/latex] number of tubes per week, is given as<\/p>\n<div id=\"fs-id1170572141865\" class=\"equation unnumbered\">[latex]p\\text{'}(x)=-0.015{e}^{-0.01x}.[\/latex]<\/div>\n<p id=\"fs-id1170571595414\">If the supermarket chain sells 100 tubes per week, what price should it set?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571543163\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571543163\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571543163\">To find the price\u2013demand equation, integrate the marginal price\u2013demand function. First find the antiderivative, then look at the particulars. Thus,<\/p>\n<div id=\"fs-id1170572216339\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ p(x)\\hfill & =\\int -0.015{e}^{-0.01x}dx\\hfill \\\\ & =-0.015\\int {e}^{-0.01x}dx.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572507579\">Using substitution, let [latex]u=-0.01x[\/latex] and [latex]du=-0.01dx.[\/latex] Then, divide both sides of the <em>du<\/em> equation by \u22120.01. This gives<\/p>\n<div class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\frac{-0.015}{-0.01}\\int {e}^{u}du\\hfill & =1.5\\int {e}^{u}du\\hfill \\\\ \\\\ & =1.5{e}^{u}+C\\hfill \\\\ & =1.5{e}^{-0.01x}+C.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572111467\">The next step is to solve for <em>C<\/em>. We know that when the price is $2.35 per tube, the demand is 50 tubes per week. This means<\/p>\n<div class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ p(50)\\hfill & =1.5{e}^{-0.01(50)}+C\\hfill \\\\ & =2.35.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170571569187\">Now, just solve for <em>C<\/em>:<\/p>\n<div id=\"fs-id1170572415392\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ C\\hfill & =2.35-1.5{e}^{-0.5}\\hfill \\\\ & =2.35-0.91\\hfill \\\\ & =1.44.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572140846\">Thus,<\/p>\n<div id=\"fs-id1170572375703\" class=\"equation unnumbered\">[latex]p(x)=1.5{e}^{-0.01x}+1.44.[\/latex]<\/div>\n<p id=\"fs-id1170571547617\">If the supermarket sells 100 tubes of toothpaste per week, the price would be<\/p>\n<div id=\"fs-id1170571653359\" class=\"equation unnumbered\">[latex]p(100)=1.5{e}^{-0.01(100)}+1.44=1.5{e}^{-1}+1.44\\approx 1.99.[\/latex]<\/div>\n<p id=\"fs-id1170572393399\">The supermarket should charge $1.99 per tube if it is selling 100 tubes per week.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571637964\" class=\"textbox examples\">\n<div id=\"fs-id1170572204890\" class=\"exercise\">\n<div class=\"textbox\">\n<h3>Evaluating a Definite Integral Involving an Exponential Function<\/h3>\n<p id=\"fs-id1170571657942\">Evaluate the definite integral [latex]{\\int }_{1}^{2}{e}^{1-x}dx.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572247799\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572247799\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572247799\">Again, substitution is the method to use. Let [latex]u=1-x,[\/latex] so [latex]du=-1dx[\/latex] or [latex]\\text{\u2212}du=dx.[\/latex] Then [latex]\\int {e}^{1-x}dx=\\text{\u2212}\\int {e}^{u}du.[\/latex] Next, change the limits of integration. Using the equation [latex]u=1-x,[\/latex] we have<\/p>\n<div id=\"fs-id1170571652053\" class=\"equation unnumbered\">[latex]\\begin{array}{c}u=1-(1)=0\\hfill \\\\ u=1-(2)=-1.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170571645662\">The integral then becomes<\/p>\n<div id=\"fs-id1170571645665\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}{\\int }_{1}^{2}{e}^{1-x}dx\\hfill & =\\text{\u2212}{\\int }_{0}^{-1}{e}^{u}du\\hfill \\\\ \\\\ \\\\ & ={\\int }_{-1}^{0}{e}^{u}du\\hfill \\\\ & ={{e}^{u}|}_{-1}^{0}\\hfill \\\\ & ={e}^{0}-({e}^{-1})\\hfill \\\\ & =\\text{\u2212}{e}^{-1}+1.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572495875\">See <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_05_06_002\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Calc_Figure_05_06_002\" class=\"wp-caption aligncenter\">\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204254\/CNX_Calc_Figure_05_06_002.jpg\" alt=\"A graph of the function f(x) = e^(1-x) over &#091;0, 3&#093;. It crosses the y axis at (0, e) as a decreasing concave up curve and symptotically approaches 0 as x goes to infinity.\" width=\"325\" height=\"208\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. The indicated area can be calculated by evaluating a definite integral using substitution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571599287\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170571599290\" class=\"exercise\">\n<div id=\"fs-id1170571599292\" class=\"textbox\">\n<p id=\"fs-id1170571599295\">Evaluate [latex]{\\int }_{0}^{2}{e}^{2x}dx.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572274760\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572274760\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572274760\">[latex]\\frac{1}{2}{\\int }_{0}^{4}{e}^{u}du=\\frac{1}{2}({e}^{4}-1)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170573206282\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170572242289\">Let [latex]u=2x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572624841\" class=\"textbox examples\">\n<h3>Growth of Bacteria in a Culture<\/h3>\n<div id=\"fs-id1170572624843\" class=\"exercise\">\n<div id=\"fs-id1170572337835\" class=\"textbox\">\n<p id=\"fs-id1170572337840\">Suppose the rate of <span class=\"no-emphasis\">growth of bacteria<\/span> in a Petri dish is given by [latex]q(t)={3}^{t},[\/latex] where [latex]t[\/latex] is given in hours and [latex]q(t)[\/latex] is given in thousands of bacteria per hour. If a culture starts with 10,000 bacteria, find a function [latex]Q(t)[\/latex] that gives the number of bacteria in the Petri dish at any time [latex]t[\/latex]. How many bacteria are in the dish after 2 hours?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571699005\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571699005\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571699005\">We have<\/p>\n<div id=\"fs-id1170571699008\" class=\"equation unnumbered\">[latex]Q(t)=\\int {3}^{t}dt=\\frac{{3}^{t}}{\\text{ln}3}+C.[\/latex]<\/div>\n<p id=\"fs-id1170571613514\">Then, at [latex]t=0[\/latex] we have [latex]Q(0)=10=\\frac{1}{\\text{ln}3}+C,[\/latex] so [latex]C\\approx 9.090[\/latex] and we get<\/p>\n<div id=\"fs-id1170571775806\" class=\"equation unnumbered\">[latex]Q(t)=\\frac{{3}^{t}}{\\text{ln}3}+9.090.[\/latex]<\/div>\n<p id=\"fs-id1170571609258\">At time [latex]t=2,[\/latex] we have<\/p>\n<div id=\"fs-id1170571698223\" class=\"equation unnumbered\">[latex]Q(2)=\\frac{{3}^{2}}{\\text{ln}3}+9.090[\/latex]<\/div>\n<div id=\"fs-id1170571637497\" class=\"equation unnumbered\">[latex]=17.282.[\/latex]<\/div>\n<p id=\"fs-id1170571637508\">After 2 hours, there are 17,282 bacteria in the dish.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571637514\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170571637518\" class=\"exercise\">\n<div id=\"fs-id1170571637520\" class=\"textbox\">\n<p id=\"fs-id1170572437454\">From <a class=\"autogenerated-content\" href=\"#fs-id1170572624841\">(Figure)<\/a>, suppose the bacteria grow at a rate of [latex]q(t)={2}^{t}.[\/latex] Assume the culture still starts with 10,000 bacteria. Find [latex]Q(t).[\/latex] How many bacteria are in the dish after 3 hours?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572444236\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572444236\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572444236\">[latex]Q(t)=\\frac{{2}^{t}}{\\text{ln}2}+8.557.[\/latex] There are 20,099 bacteria in the dish after 3 hours.<\/p>\n<\/div>\n<div id=\"fs-id1170573623952\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170572444227\">Use the procedure from <a class=\"autogenerated-content\" href=\"#fs-id1170572624841\">(Figure)<\/a> to solve the problem.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571807209\" class=\"textbox examples\">\n<h3>Fruit Fly Population Growth<\/h3>\n<div id=\"fs-id1170571807211\" class=\"exercise\">\n<div id=\"fs-id1170571807213\" class=\"textbox\">\n<p id=\"fs-id1170571807219\">Suppose a population of <span class=\"no-emphasis\">fruit flies<\/span> increases at a rate of [latex]g(t)=2{e}^{0.02t},[\/latex] in flies per day. If the initial population of fruit flies is 100 flies, how many flies are in the population after 10 days?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572183855\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572183855\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572183855\">Let [latex]G(t)[\/latex] represent the number of flies in the population at time [latex]t[\/latex]. Applying the net change theorem, we have<\/p>\n<div id=\"fs-id1170571712220\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ G(10)\\hfill & =G(0)+{\\int }_{0}^{10}2{e}^{0.02t}dt\\hfill \\\\ & =100+{\\left[\\frac{2}{0.02}{e}^{0.02t}\\right]|}_{0}^{10}\\hfill \\\\ & =100+{\\left[100{e}^{0.02t}\\right]|}_{0}^{10}\\hfill \\\\ & =100+100{e}^{0.2}-100\\hfill \\\\ & \\approx 122.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572338467\">There are 122 flies in the population after 10 days.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572338474\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572338477\" class=\"exercise\">\n<div id=\"fs-id1170572338479\" class=\"textbox\">\n<p id=\"fs-id1170572338481\">Suppose the rate of growth of the fly population is given by [latex]g(t)={e}^{0.01t},[\/latex] and the initial fly population is 100 flies. How many flies are in the population after 15 days?<\/p>\n<\/div>\n<div id=\"fs-id1170571653080\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571653080\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571653080\" class=\"hidden-answer\" style=\"display: none\">There are 116 flies.<\/div>\n<div id=\"fs-id1170573548872\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170571653073\">Use the process from <a class=\"autogenerated-content\" href=\"#fs-id1170571807209\">(Figure)<\/a> to solve the problem.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1170571653091\" class=\"exercise\">\n<div id=\"fs-id1170571653093\" class=\"textbox\">\n<h3>Evaluating a Definite Integral Using Substitution<\/h3>\n<p id=\"fs-id1170572217480\">Evaluate the definite integral using substitution: [latex]{\\int }_{1}^{2}\\frac{{e}^{1\\text{\/}x}}{{x}^{2}}dx.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572587719\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572587719\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572587719\">This problem requires some rewriting to simplify applying the properties. First, rewrite the exponent on [latex]e[\/latex] as a power of [latex]x[\/latex], then bring the [latex]x[\/latex]<sup>2<\/sup> in the denominator up to the numerator using a negative exponent. We have<\/p>\n<div id=\"fs-id1170572587738\" class=\"equation unnumbered\">[latex]{\\int }_{1}^{2}\\frac{{e}^{1\\text{\/}x}}{{x}^{2}}dx={\\int }_{1}^{2}{e}^{{x}^{-1}}{x}^{-2}dx.[\/latex]<\/div>\n<p id=\"fs-id1170571636291\">Let [latex]u={x}^{-1},[\/latex] the exponent on [latex]e[\/latex]. Then<\/p>\n<div id=\"fs-id1170571636314\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill du& =\\text{\u2212}{x}^{-2}dx\\hfill \\\\ \\hfill -du& ={x}^{-2}dx.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170571599633\">Bringing the negative sign outside the integral sign, the problem now reads<\/p>\n<div class=\"equation unnumbered\">[latex]\\text{\u2212}\\int {e}^{u}du.[\/latex]<\/div>\n<p id=\"fs-id1170571599663\">Next, change the limits of integration:<\/p>\n<div id=\"fs-id1170571599666\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ u={(1)}^{-1}=1\\hfill \\\\ u={(2)}^{-1}=\\frac{1}{2}.\\hfill \\end{array}[\/latex]<\/div>\n<p>Notice that now the limits begin with the larger number, meaning we must multiply by \u22121 and interchange the limits. Thus,<\/p>\n<div id=\"fs-id1170572293466\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ \\\\ \\text{\u2212}{\\int }_{1}^{1\\text{\/}2}{e}^{u}du\\hfill & ={\\int }_{1\\text{\/}2}^{1}{e}^{u}du\\hfill \\\\ & ={e}^{u}{|}_{1\\text{\/}2}^{1}\\hfill \\\\ & =e-{e}^{1\\text{\/}2}\\hfill \\\\ & =e-\\sqrt{e}.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572554374\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572554378\" class=\"exercise\">\n<div id=\"fs-id1170572554380\" class=\"textbox\">\n<p id=\"fs-id1170572554382\">Evaluate the definite integral using substitution: [latex]{\\int }_{1}^{2}\\frac{1}{{x}^{3}}{e}^{4{x}^{-2}}dx.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572628419\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572628419\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572628419\">[latex]{\\int }_{1}^{2}\\frac{1}{{x}^{3}}{e}^{4{x}^{-2}}dx=\\frac{1}{8}\\left[{e}^{4}-e\\right][\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170571419835\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170572554429\">Let [latex]u=4{x}^{-2}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572565328\" class=\"bc-section section\">\n<h1>Integrals Involving Logarithmic Functions<\/h1>\n<p id=\"fs-id1170572565333\">Integrating functions of the form [latex]f(x)={x}^{-1}[\/latex] result in the absolute value of the natural log function, as shown in the following rule. Integral formulas for other logarithmic functions, such as [latex]f(x)=\\text{ln}x[\/latex] and [latex]f(x)={\\text{log}}_{a}x,[\/latex] are also included in the rule.<\/p>\n<div id=\"fs-id1170572543658\" class=\"textbox key-takeaways\">\n<h3>Rule: Integration Formulas Involving Logarithmic Functions<\/h3>\n<p id=\"fs-id1170572543663\">The following formulas can be used to evaluate integrals involving logarithmic functions.<\/p>\n<div id=\"fs-id1170572543666\" class=\"equation\">[latex]\\begin{array}{ccc}\\hfill \\int {x}^{-1}dx& =\\hfill & \\text{ln}|x|+C\\hfill \\\\ \\hfill \\int \\text{ln}xdx& =\\hfill & x\\text{ln}x-x+C=x(\\text{ln}x-1)+C\\hfill \\\\ \\hfill \\int {\\text{log}}_{a}xdx& =\\hfill & \\frac{x}{\\text{ln}a}(\\text{ln}x-1)+C\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1170571653392\" class=\"textbox examples\">\n<h3>Finding an Antiderivative Involving [latex]\\text{ln}x[\/latex]<\/h3>\n<div id=\"fs-id1170571653394\" class=\"exercise\">\n<div id=\"fs-id1170571653396\" class=\"textbox\">\n<p id=\"fs-id1170571653410\">Find the antiderivative of the function [latex]\\frac{3}{x-10}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571653435\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571653435\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571653435\">First factor the 3 outside the integral symbol. Then use the [latex]u[\/latex]<sup>\u22121<\/sup> rule. Thus,<\/p>\n<div id=\"fs-id1170571653444\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}\\int \\frac{3}{x-10}dx\\hfill & =3\\int \\frac{1}{x-10}dx\\hfill \\\\ \\\\ \\\\ & =3\\int \\frac{du}{u}\\hfill \\\\ & =3\\text{ln}|u|+C\\hfill \\\\ & =3\\text{ln}|x-10|+C,x\\ne 10.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170571597433\">See <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_05_06_003\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Calc_Figure_05_06_003\" class=\"wp-caption aligncenter\">\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204256\/CNX_Calc_Figure_05_06_003.jpg\" alt=\"A graph of the function f(x) = 3 \/ (x \u2013 10). There is an asymptote at x=10. The first segment is a decreasing concave down curve that approaches 0 as x goes to negative infinity and approaches negative infinity as x goes to 10. The second segment is a decreasing concave up curve that approaches infinity as x goes to 10 and approaches 0 as x approaches infinity.\" width=\"325\" height=\"246\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. The domain of this function is [latex]x\\ne 10.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571597473\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170571597476\" class=\"exercise\">\n<div id=\"fs-id1170571597479\" class=\"textbox\">\n<p id=\"fs-id1170571597481\">Find the antiderivative of [latex]\\frac{1}{x+2}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572480516\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572480516\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572480516\">[latex]\\text{ln}|x+2|+C[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170573721513\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170572480506\">Follow the pattern from <a class=\"autogenerated-content\" href=\"#fs-id1170571653392\">(Figure)<\/a> to solve the problem.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572480542\" class=\"textbox examples\">\n<h3>Finding an Antiderivative of a Rational Function<\/h3>\n<div id=\"fs-id1170572480545\" class=\"exercise\">\n<div id=\"fs-id1170572480547\" class=\"textbox\">\n<p id=\"fs-id1170572480552\">Find the antiderivative of [latex]\\frac{2{x}^{3}+3x}{{x}^{4}+3{x}^{2}}.[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>This can be rewritten as [latex]\\int (2{x}^{3}+3x){({x}^{4}+3{x}^{2})}^{-1}dx.[\/latex] Use substitution. Let [latex]u={x}^{4}+3{x}^{2},[\/latex] then [latex]du=4{x}^{3}+6x.[\/latex] Alter <em>du<\/em> by factoring out the 2. Thus,<\/p>\n<div id=\"fs-id1170571807881\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\hfill du& =\\hfill & (4{x}^{3}+6x)dx\\hfill \\\\ & =\\hfill & 2(2{x}^{3}+3x)dx\\hfill \\\\ \\hfill \\frac{1}{2}du& =\\hfill & (2{x}^{3}+3x)dx.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572373464\">Rewrite the integrand in [latex]u[\/latex]:<\/p>\n<div id=\"fs-id1170572331833\" class=\"equation unnumbered\">[latex]\\int (2{x}^{3}+3x){({x}^{4}+3{x}^{2})}^{-1}dx=\\frac{1}{2}\\int {u}^{-1}du.[\/latex]<\/div>\n<p id=\"fs-id1170571733965\">Then we have<\/p>\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ll}\\frac{1}{2}\\int {u}^{-1}du\\hfill & =\\frac{1}{2}\\text{ln}|u|+C\\hfill \\\\ \\\\ & =\\frac{1}{2}\\text{ln}|{x}^{4}+3{x}^{2}|+C.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571649922\" class=\"textbox examples\">\n<div class=\"exercise\">\n<div class=\"textbox\">\n<h3>Finding an Antiderivative of a Logarithmic Function<\/h3>\n<p id=\"fs-id1170571649932\">Find the antiderivative of the log function [latex]{\\text{log}}_{2}x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571649954\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571649954\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571649954\">Follow the format in the formula listed in the rule on integration formulas involving logarithmic functions. Based on this format, we have<\/p>\n<div id=\"fs-id1170571649959\" class=\"equation unnumbered\">[latex]\\int {\\text{log}}_{2}xdx=\\frac{x}{\\text{ln}2}(\\text{ln}x-1)+C.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571660181\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170571660184\" class=\"exercise\">\n<div id=\"fs-id1170571660186\" class=\"textbox\">\n<p id=\"fs-id1170571660188\">Find the antiderivative of [latex]{\\text{log}}_{3}x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571660214\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571660214\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571660214\">[latex]\\frac{x}{\\text{ln}3}(\\text{ln}x-1)+C[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170573418990\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170571660204\">Follow <a class=\"autogenerated-content\" href=\"#fs-id1170571649922\">(Figure)<\/a> and refer to the rule on integration formulas involving logarithmic functions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572274900\"><a class=\"autogenerated-content\" href=\"#fs-id1170572274908\">(Figure)<\/a> is a definite integral of a trigonometric function. With trigonometric functions, we often have to apply a trigonometric property or an identity before we can move forward. Finding the right form of the integrand is usually the key to a smooth integration.<\/p>\n<div id=\"fs-id1170572274908\" class=\"textbox examples\">\n<h3>Evaluating a Definite Integral<\/h3>\n<div id=\"fs-id1170572274910\" class=\"exercise\">\n<div id=\"fs-id1170572274912\" class=\"textbox\">\n<p id=\"fs-id1170572274917\">Find the definite integral of [latex]{\\int }_{0}^{\\pi \\text{\/}2}\\frac{ \\sin x}{1+ \\cos x}dx.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571636148\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571636148\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571636148\">We need substitution to evaluate this problem. Let [latex]u=1+ \\cos x,,[\/latex] so [latex]du=\\text{\u2212} \\sin xdx.[\/latex] Rewrite the integral in terms of [latex]u[\/latex], changing the limits of integration as well. Thus,<\/p>\n<div id=\"fs-id1170571636203\" class=\"equation unnumbered\">[latex]\\begin{array}{c}u=1+ \\cos (0)=2\\hfill \\\\ u=1+ \\cos (\\frac{\\pi }{2})=1.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572510108\">Then<\/p>\n<div class=\"equation unnumbered\">[latex]\\begin{array}{cc}{\\int }_{0}^{\\pi \\text{\/}2}\\frac{ \\sin x}{1+ \\cos x}\\hfill & =\\text{\u2212}{\\int }_{2}^{1}{u}^{-1}du\\hfill \\\\ \\\\ \\\\ & ={\\int }_{1}^{2}{u}^{-1}du\\hfill \\\\ & ={\\text{ln}|u||}_{1}^{2}\\hfill \\\\ & =\\left[\\text{ln}2-\\text{ln}1\\right]\\hfill \\\\ & =\\text{ln}2.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571712566\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1170571712573\">\n<li>Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay.<\/li>\n<li>Substitution is often used to evaluate integrals involving exponential functions or logarithms.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170571712590\" class=\"key-equations\">\n<h1>Key Equations<\/h1>\n<ul>\n<li><strong>Integrals of Exponential Functions<\/strong><br \/>\n[latex]\\int {e}^{x}dx={e}^{x}+C[\/latex]<br \/>\n[latex]\\int {a}^{x}dx=\\frac{{a}^{x}}{\\text{ln}a}+C[\/latex]<\/li>\n<li><strong>Integration Formulas Involving Logarithmic Functions<\/strong><br \/>\n[latex]\\int {x}^{-1}dx=\\text{ln}|x|+C[\/latex]<br \/>\n[latex]\\int \\text{ln}xdx=x\\text{ln}x-x+C=x(\\text{ln}x-1)+C[\/latex]<br \/>\n[latex]\\int {\\text{log}}_{a}xdx=\\frac{x}{\\text{ln}a}(\\text{ln}x-1)+C[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170571699478\" class=\"textbox exercises\">\n<p id=\"fs-id1170571699482\">In the following exercises, compute each indefinite integral.<\/p>\n<div id=\"fs-id1170571699485\" class=\"exercise\">\n<div id=\"fs-id1170571699487\" class=\"textbox\">\n<p id=\"fs-id1170571699489\">[latex]\\int {e}^{2x}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571678838\" class=\"exercise\">\n<div id=\"fs-id1170571678840\" class=\"textbox\">\n<p id=\"fs-id1170571678842\">[latex]\\int {e}^{-3x}dx[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571678870\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571678870\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571678870\">[latex]\\frac{-1}{3}{e}^{-3x}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1170572601243\" class=\"textbox\">\n<p id=\"fs-id1170572601245\">[latex]\\int {2}^{x}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572601296\" class=\"exercise\">\n<div id=\"fs-id1170572601298\" class=\"textbox\">\n<p id=\"fs-id1170572601300\">[latex]\\int {3}^{\\text{\u2212}x}dx[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571679785\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571679785\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571679785\">[latex]-\\frac{{3}^{\\text{\u2212}x}}{\\text{ln}3}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571679817\" class=\"exercise\">\n<div id=\"fs-id1170571679819\" class=\"textbox\">\n<p id=\"fs-id1170571679821\">[latex]\\int \\frac{1}{2x}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572611839\" class=\"exercise\">\n<div id=\"fs-id1170572611841\" class=\"textbox\">\n<p id=\"fs-id1170572611843\">[latex]\\int \\frac{2}{x}dx[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572611870\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572611870\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572611870\">[latex]\\text{ln}({x}^{2})+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572611894\" class=\"exercise\">\n<div id=\"fs-id1170572611896\" class=\"textbox\">\n<p>[latex]\\int \\frac{1}{{x}^{2}}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572333073\" class=\"exercise\">\n<div id=\"fs-id1170572333075\" class=\"textbox\">\n<p id=\"fs-id1170572333077\">[latex]\\int \\frac{1}{\\sqrt{x}}dx[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572333105\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572333105\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572333105\">[latex]2\\sqrt{x}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572333121\">In the following exercises, find each indefinite integral by using appropriate substitutions.<\/p>\n<div id=\"fs-id1170572333125\" class=\"exercise\">\n<div id=\"fs-id1170572333127\" class=\"textbox\">\n<p id=\"fs-id1170572333129\">[latex]\\int \\frac{\\text{ln}x}{x}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571638228\" class=\"exercise\">\n<div id=\"fs-id1170571638230\" class=\"textbox\">\n<p id=\"fs-id1170571638233\">[latex]\\int \\frac{dx}{x{(\\text{ln}x)}^{2}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571810853\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571810853\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571810853\">[latex]-\\frac{1}{\\text{ln}x}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571810876\" class=\"exercise\">\n<div id=\"fs-id1170571810878\" class=\"textbox\">\n<p id=\"fs-id1170571810880\">[latex]\\int \\frac{dx}{x\\text{ln}x}(x>1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571697166\" class=\"exercise\">\n<div id=\"fs-id1170571697168\" class=\"textbox\">\n<p id=\"fs-id1170571697170\">[latex]\\int \\frac{dx}{x\\text{ln}x\\text{ln}(\\text{ln}x)}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571697227\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571697227\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571697227\">[latex]\\text{ln}(\\text{ln}(\\text{ln}x))+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571539147\" class=\"exercise\">\n<div id=\"fs-id1170571539149\" class=\"textbox\">\n<p id=\"fs-id1170571539151\">[latex]\\int \\tan \\theta d\\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571539202\" class=\"exercise\">\n<div id=\"fs-id1170571539204\" class=\"textbox\">\n<p id=\"fs-id1170571539206\">[latex]\\int \\frac{ \\cos x-x \\sin x}{x \\cos x}dx[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170571543234\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571543234\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571543234\" class=\"hidden-answer\" style=\"display: none\">[latex]\\text{ln}(x \\cos x)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571543268\" class=\"exercise\">\n<div id=\"fs-id1170571543270\" class=\"textbox\">\n<p id=\"fs-id1170571543272\">[latex]\\int \\frac{\\text{ln}( \\sin x)}{ \\tan x}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572456410\" class=\"exercise\">\n<div id=\"fs-id1170572456412\" class=\"textbox\">\n<p id=\"fs-id1170572456414\">[latex]\\int \\text{ln}( \\cos x) \\tan xdx[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>[latex]-\\frac{1}{2}{(\\text{ln}( \\cos (x)))}^{2}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572396562\" class=\"exercise\">\n<div id=\"fs-id1170572396564\" class=\"textbox\">\n<p id=\"fs-id1170572396567\">[latex]\\int x{e}^{\\text{\u2212}{x}^{2}}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1170571580982\">[latex]\\int {x}^{2}{e}^{\\text{\u2212}{x}^{3}}dx[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572218578\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572218578\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572218578\">[latex]\\frac{\\text{\u2212}{e}^{\\text{\u2212}{x}^{3}}}{3}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218606\" class=\"exercise\">\n<div id=\"fs-id1170572218608\" class=\"textbox\">\n<p id=\"fs-id1170572218611\">[latex]\\int {e}^{ \\sin x} \\cos xdx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218671\" class=\"exercise\">\n<div class=\"textbox\">\n<p>[latex]\\int {e}^{ \\tan x}{ \\sec }^{2}xdx[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572386146\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572386146\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572386146\">[latex]{e}^{ \\tan x}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572386168\" class=\"exercise\">\n<div id=\"fs-id1170572386170\" class=\"textbox\">\n<p id=\"fs-id1170572386172\">[latex]\\int {e}^{\\text{ln}x}\\frac{dx}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p>[latex]\\int \\frac{{e}^{\\text{ln}(1-t)}}{1-t}dt[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572643259\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572643259\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572643259\">[latex]t+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following exercises, verify by differentiation that [latex]\\int \\text{ln}xdx=x(\\text{ln}x-1)+C,[\/latex] then use appropriate changes of variables to compute the integral.<\/p>\n<div id=\"fs-id1170572373699\" class=\"exercise\">\n<div class=\"textbox\">\n<p>[latex]\\int \\text{ln}xdx[\/latex][latex](Hint\\text{:}\\int \\text{ln}xdx=\\frac{1}{2}\\int x\\text{ln}({x}^{2})dx)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571710720\" class=\"exercise\">\n<div id=\"fs-id1170571710722\" class=\"textbox\">\n<p id=\"fs-id1170571710724\">[latex]\\int {x}^{2}{\\text{ln}}^{2}xdx[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572399000\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572399000\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572399000\">[latex]\\frac{1}{9}{x}^{3}(\\text{ln}({x}^{3})-1)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572399045\" class=\"exercise\">\n<div id=\"fs-id1170572399048\" class=\"textbox\">\n<p id=\"fs-id1170572399050\">[latex]\\int \\frac{\\text{ln}x}{{x}^{2}}dx[\/latex][latex](Hint\\text{:}\\text{Set}u=\\frac{1}{x}\\text{.})[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572415189\" class=\"exercise\">\n<div id=\"fs-id1170572415191\" class=\"textbox\">\n<p id=\"fs-id1170572415193\">[latex]\\int \\frac{\\text{ln}x}{\\sqrt{x}}dx[\/latex][latex](Hint\\text{:}\\text{Set}u=\\sqrt{x}\\text{.})[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572168712\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572168712\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572168712\">[latex]2\\sqrt{x}(\\text{ln}x-2)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1170572168748\" class=\"textbox\">\n<p id=\"fs-id1170572168750\">Write an integral to express the area under the graph of [latex]y=\\frac{1}{t}[\/latex] from [latex]t=1[\/latex] to <em>e<sup>x<\/sup><\/em> and evaluate the integral.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572629242\" class=\"exercise\">\n<div id=\"fs-id1170572629244\" class=\"textbox\">\n<p id=\"fs-id1170572629246\">Write an integral to express the area under the graph of [latex]y={e}^{t}[\/latex] between [latex]t=0[\/latex] and [latex]t=\\text{ln}x,[\/latex] and evaluate the integral.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571689752\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571689752\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571689752\">[latex]{\\int }_{0}^{\\text{ln}x}{e}^{t}dt={e}^{t}{|}_{0}^{\\text{ln}x}={e}^{\\text{ln}x}-{e}^{0}=x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572569960\">In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.<\/p>\n<div id=\"fs-id1170572569965\" class=\"exercise\">\n<div id=\"fs-id1170572569967\" class=\"textbox\">\n<p id=\"fs-id1170572569969\">[latex]\\int \\tan (2x)dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571571932\" class=\"exercise\">\n<div id=\"fs-id1170571571934\" class=\"textbox\">\n<p id=\"fs-id1170571571936\">[latex]\\int \\frac{ \\sin (3x)- \\cos (3x)}{ \\sin (3x)+ \\cos (3x)}dx[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571572016\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571572016\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571572016\">[latex]-\\frac{1}{3}\\text{ln}( \\sin (3x)+ \\cos (3x))[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572379018\" class=\"exercise\">\n<div id=\"fs-id1170572379020\" class=\"textbox\">\n<p id=\"fs-id1170572379022\">[latex]\\int \\frac{x \\sin ({x}^{2})}{ \\cos ({x}^{2})}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571777887\" class=\"exercise\">\n<div id=\"fs-id1170571777889\" class=\"textbox\">\n<p id=\"fs-id1170571777891\">[latex]\\int x \\csc ({x}^{2})dx[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571777931\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571777931\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571777931\">[latex]-\\frac{1}{2}\\text{ln}| \\csc ({x}^{2})+ \\cot ({x}^{2})|+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572310019\" class=\"exercise\">\n<div id=\"fs-id1170572310021\" class=\"textbox\">\n<p id=\"fs-id1170572310023\">[latex]\\int \\text{ln}( \\cos x) \\tan xdx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571613744\" class=\"exercise\">\n<div id=\"fs-id1170571613746\" class=\"textbox\">\n<p id=\"fs-id1170571613748\">[latex]\\int \\text{ln}( \\csc x) \\cot xdx[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571613792\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571613792\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571613792\">[latex]-\\frac{1}{2}{(\\text{ln}( \\csc x))}^{2}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572451363\" class=\"exercise\">\n<div id=\"fs-id1170572451366\" class=\"textbox\">\n<p id=\"fs-id1170572451368\">[latex]\\int \\frac{{e}^{x}-{e}^{\\text{\u2212}x}}{{e}^{x}+{e}^{\\text{\u2212}x}}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572346997\">In the following exercises, evaluate the definite integral.<\/p>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1170572347004\">[latex]{\\int }_{1}^{2}\\frac{1+2x+{x}^{2}}{3x+3{x}^{2}+{x}^{3}}dx[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170572347069\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572347069\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572347069\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{1}{3}\\text{ln}(\\frac{26}{7})[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572329945\" class=\"exercise\">\n<div id=\"fs-id1170572329947\" class=\"textbox\">\n<p id=\"fs-id1170572329949\">[latex]{\\int }_{0}^{\\pi \\text{\/}4} \\tan xdx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572330008\" class=\"exercise\">\n<div id=\"fs-id1170572330010\" class=\"textbox\">\n<p id=\"fs-id1170572330012\">[latex]{\\int }_{0}^{\\pi \\text{\/}3}\\frac{ \\sin x- \\cos x}{ \\sin x+ \\cos x}dx[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571613655\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571613655\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571613655\">[latex]\\text{ln}(\\sqrt{3}-1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571613677\" class=\"exercise\">\n<div id=\"fs-id1170571613679\" class=\"textbox\">\n<p id=\"fs-id1170571613681\">[latex]{\\int }_{\\pi \\text{\/}6}^{\\pi \\text{\/}2} \\csc xdx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p>[latex]{\\int }_{\\pi \\text{\/}4}^{\\pi \\text{\/}3} \\cot xdx[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572412258\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572412258\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572412258\">[latex]\\frac{1}{2}\\text{ln}\\frac{3}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572412280\">In the following exercises, integrate using the indicated substitution.<\/p>\n<div id=\"fs-id1170572412283\" class=\"exercise\">\n<div id=\"fs-id1170572412285\" class=\"textbox\">\n<p id=\"fs-id1170572412287\">[latex]\\int \\frac{x}{x-100}dx;u=x-100[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572627084\" class=\"exercise\">\n<div id=\"fs-id1170572627086\" class=\"textbox\">\n<p id=\"fs-id1170572627088\">[latex]\\int \\frac{y-1}{y+1}dy;u=y+1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572627138\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572627138\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572627138\">[latex]y-2\\text{ln}|y+1|+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572627173\" class=\"exercise\">\n<div id=\"fs-id1170572627176\" class=\"textbox\">\n<p id=\"fs-id1170572572238\">[latex]\\int \\frac{1-{x}^{2}}{3x-{x}^{3}}dx;u=3x-{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572572338\" class=\"exercise\">\n<div id=\"fs-id1170572572340\" class=\"textbox\">\n<p>[latex]\\int \\frac{ \\sin x+ \\cos x}{ \\sin x- \\cos x}dx;u= \\sin x- \\cos x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571712850\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571712850\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571712850\">[latex]\\text{ln}| \\sin x- \\cos x|+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572129747\" class=\"exercise\">\n<div id=\"fs-id1170572129749\" class=\"textbox\">\n<p id=\"fs-id1170572129752\">[latex]\\int {e}^{2x}\\sqrt{1-{e}^{2x}}dx;u={e}^{2x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572379189\" class=\"exercise\">\n<div id=\"fs-id1170572379191\" class=\"textbox\">\n<p id=\"fs-id1170572379193\">[latex]\\int \\text{ln}(x)\\frac{\\sqrt{1-{(\\text{ln}x)}^{2}}}{x}dx;u=\\text{ln}x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572379266\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572379266\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572379266\">[latex]-\\frac{1}{3}{(1-(\\text{ln}{x}^{2}))}^{3\\text{\/}2}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572330224\">In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate <em>R<\/em><sub>50<\/sub> and solve for the exact area.<\/p>\n<div id=\"fs-id1170572330236\" class=\"exercise\">\n<div id=\"fs-id1170572330238\" class=\"textbox\">\n<p id=\"fs-id1170572330240\"><strong>[T]<\/strong>[latex]y={e}^{x}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572129062\" class=\"exercise\">\n<div id=\"fs-id1170572129064\" class=\"textbox\">\n<p id=\"fs-id1170572129066\"><strong>[T]<\/strong>[latex]y={e}^{\\text{\u2212}x}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572129108\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572129108\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572129108\">Exact solution: [latex]\\frac{e-1}{e},{R}_{50}=0.6258.[\/latex] Since [latex]f[\/latex] is decreasing, the right endpoint estimate underestimates the area.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572129147\" class=\"exercise\">\n<div id=\"fs-id1170572129149\" class=\"textbox\">\n<p id=\"fs-id1170572129152\"><strong>[T]<\/strong>[latex]y=\\text{ln}(x)[\/latex] over [latex]\\left[1,2\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572223556\" class=\"exercise\">\n<div id=\"fs-id1170572223558\" class=\"textbox\">\n<p id=\"fs-id1170572223560\"><strong>[T]<\/strong>[latex]y=\\frac{x+1}{{x}^{2}+2x+6}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571568951\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571568951\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571568951\">Exact solution: [latex]\\frac{2\\text{ln}(3)-\\text{ln}(6)}{2},{R}_{50}=0.2033.[\/latex] Since [latex]f[\/latex] is increasing, the right endpoint estimate overestimates the area.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571569011\" class=\"exercise\">\n<div id=\"fs-id1170571569013\" class=\"textbox\">\n<p id=\"fs-id1170571569015\"><strong>[T]<\/strong>[latex]y={2}^{x}[\/latex] over [latex]\\left[-1,0\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572309777\" class=\"exercise\">\n<div id=\"fs-id1170572309779\" class=\"textbox\">\n<p id=\"fs-id1170572309781\"><strong>[T]<\/strong>[latex]y=\\text{\u2212}{2}^{\\text{\u2212}x}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571628910\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571628910\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571628910\">Exact solution: [latex]-\\frac{1}{\\text{ln}(4)},{R}_{50}=-0.7164.[\/latex] Since [latex]f[\/latex] is increasing, the right endpoint estimate overestimates the area (the actual area is a larger negative number).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170571628956\">In the following exercises, [latex]f(x)\\ge 0[\/latex] for [latex]a\\le x\\le b.[\/latex] Find the area under the graph of [latex]f(x)[\/latex] between the given values [latex]a[\/latex] and [latex]b[\/latex] by integrating.<\/p>\n<div id=\"fs-id1170572582554\" class=\"exercise\">\n<div id=\"fs-id1170572582556\" class=\"textbox\">\n<p id=\"fs-id1170572582558\">[latex]f(x)=\\frac{{\\text{log}}_{10}(x)}{x};a=10,b=100[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572582636\" class=\"exercise\">\n<div id=\"fs-id1170572582638\" class=\"textbox\">\n<p id=\"fs-id1170572582641\">[latex]f(x)=\\frac{{\\text{log}}_{2}(x)}{x};a=32,b=64[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572351521\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572351521\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572351521\">[latex]\\frac{11}{2}\\text{ln}2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572351541\" class=\"exercise\">\n<div id=\"fs-id1170572351543\" class=\"textbox\">\n<p id=\"fs-id1170572351545\">[latex]f(x)={2}^{\\text{\u2212}x};a=1,b=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571638055\" class=\"exercise\">\n<div id=\"fs-id1170571638057\" class=\"textbox\">\n<p id=\"fs-id1170571638060\">[latex]f(x)={2}^{\\text{\u2212}x};a=3,b=4[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571638104\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571638104\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571638104\">[latex]\\frac{1}{\\text{ln}(65,536)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571638130\" class=\"exercise\">\n<div id=\"fs-id1170571638132\" class=\"textbox\">\n<p id=\"fs-id1170571638134\">Find the area under the graph of the function [latex]f(x)=x{e}^{\\text{\u2212}{x}^{2}}[\/latex] between [latex]x=0[\/latex] and [latex]x=5.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571610335\" class=\"exercise\">\n<div id=\"fs-id1170571610337\" class=\"textbox\">\n<p id=\"fs-id1170571610339\">Compute the integral of [latex]f(x)=x{e}^{\\text{\u2212}{x}^{2}}[\/latex] and find the smallest value of <em>N<\/em> such that the area under the graph [latex]f(x)=x{e}^{\\text{\u2212}{x}^{2}}[\/latex] between [latex]x=N[\/latex] and [latex]x=N+10[\/latex] is, at most, 0.01.<\/p>\n<\/div>\n<div id=\"fs-id1170572504510\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572504510\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572504510\" class=\"hidden-answer\" style=\"display: none\">[latex]{\\int }_{N}^{N+1}x{e}^{\\text{\u2212}{x}^{2}}dx=\\frac{1}{2}({e}^{\\text{\u2212}{N}^{2}}-{e}^{\\text{\u2212}{(N+1)}^{2}}).[\/latex] The quantity is less than 0.01 when [latex]N=2.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572368492\" class=\"exercise\">\n<div id=\"fs-id1170572368494\" class=\"textbox\">\n<p id=\"fs-id1170572368496\">Find the limit, as <em>N<\/em> tends to infinity, of the area under the graph of [latex]f(x)=x{e}^{\\text{\u2212}{x}^{2}}[\/latex] between [latex]x=0[\/latex] and [latex]x=5.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572448465\" class=\"exercise\">\n<div id=\"fs-id1170572448467\" class=\"textbox\">\n<p id=\"fs-id1170572448469\">Show that [latex]{\\int }_{a}^{b}\\frac{dt}{t}={\\int }_{1\\text{\/}b}^{1\\text{\/}a}\\frac{dt}{t}[\/latex] when [latex]0<a\\le b.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572306386\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572306386\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572306386\">[latex]{\\int }_{a}^{b}\\frac{dx}{x}=\\text{ln}(b)-\\text{ln}(a)=\\text{ln}(\\frac{1}{a})-\\text{ln}(\\frac{1}{b})={\\int }_{1\\text{\/}b}^{1\\text{\/}a}\\frac{dx}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572503285\" class=\"exercise\">\n<div id=\"fs-id1170572503288\" class=\"textbox\">\n<p id=\"fs-id1170572503290\">Suppose that [latex]f(x)>0[\/latex] for all [latex]x[\/latex] and that [latex]f[\/latex] and [latex]g[\/latex] are differentiable. Use the identity [latex]{f}^{g}={e}^{g\\text{ln}f}[\/latex] and the chain rule to find the derivative of [latex]{f}^{g}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1170572558121\" class=\"textbox\">\n<p id=\"fs-id1170572558123\">Use the previous exercise to find the antiderivative of [latex]h(x)={x}^{x}(1+\\text{ln}x)[\/latex] and evaluate [latex]{\\int }_{2}^{3}{x}^{x}(1+\\text{ln}x)dx.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571814004\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571814004\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571814004\">23<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571814009\" class=\"exercise\">\n<div id=\"fs-id1170571814011\" class=\"textbox\">\n<p id=\"fs-id1170571814013\">Show that if [latex]c>0,[\/latex] then the integral of [latex]1\\text{\/}x[\/latex] from <em>ac<\/em> to <em>bc<\/em> [latex](0<a<b)[\/latex] is the same as the integral of [latex]1\\text{\/}x[\/latex] from [latex]a[\/latex] to [latex]b[\/latex].<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572307624\">The following exercises are intended to derive the fundamental properties of the natural log starting from the <em>Definition\/em&gt; [latex]\\text{ln}(x)={\\int }_{1}^{x}\\frac{dt}{t},[\/latex] using properties of the definite integral and making no further assumptions.<\/em><\/p>\n<div id=\"fs-id1170572307674\" class=\"exercise\">\n<div id=\"fs-id1170572307676\" class=\"textbox\">\n<p id=\"fs-id1170572307678\">Use the identity [latex]\\text{ln}(x)={\\int }_{1}^{x}\\frac{dt}{t}[\/latex] to derive the identity [latex]\\text{ln}(\\frac{1}{x})=\\text{\u2212}\\text{ln}x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572296602\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572296602\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572296602\">We may assume that [latex]x>1,\\text{so}\\frac{1}{x}<1.[\/latex] Then, [latex]{\\int }_{1}^{1\\text{\/}x}\\frac{dt}{t}.[\/latex] Now make the substitution [latex]u=\\frac{1}{t},[\/latex] so [latex]du=-\\frac{dt}{{t}^{2}}[\/latex] and [latex]\\frac{du}{u}=-\\frac{dt}{t},[\/latex] and change endpoints: [latex]{\\int }_{1}^{1\\text{\/}x}\\frac{dt}{t}=\\text{\u2212}{\\int }_{1}^{x}\\frac{du}{u}=\\text{\u2212}\\text{ln}x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p>Use a change of variable in the integral [latex]{\\int }_{1}^{xy}\\frac{1}{t}dt[\/latex] to show that [latex]\\text{ln}xy=\\text{ln}x+\\text{ln}y\\text{ for }x,y>0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571629695\" class=\"exercise\">\n<div id=\"fs-id1170571629697\" class=\"textbox\">\n<p id=\"fs-id1170571629699\">Use the identity [latex]\\text{ln}x={\\int }_{1}^{x}\\frac{dt}{x}[\/latex] to show that [latex]\\text{ln}(x)[\/latex] is an increasing function of [latex]x[\/latex] on [latex]\\left[0,\\infty ),[\/latex] and use the previous exercises to show that the range of [latex]\\text{ln}(x)[\/latex] is [latex](\\text{\u2212}\\infty ,\\infty ).[\/latex] Without any further assumptions, conclude that [latex]\\text{ln}(x)[\/latex] has an inverse function defined on [latex](\\text{\u2212}\\infty ,\\infty ).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571624156\" class=\"exercise\">\n<div id=\"fs-id1170571624158\" class=\"textbox\">\n<p id=\"fs-id1170571624160\">Pretend, for the moment, that we do not know that [latex]{e}^{x}[\/latex] is the inverse function of [latex]\\text{ln}(x),[\/latex] but keep in mind that [latex]\\text{ln}(x)[\/latex] has an inverse function defined on [latex](\\text{\u2212}\\infty ,\\infty ).[\/latex] Call it <em>E<\/em>. Use the identity [latex]\\text{ln}xy=\\text{ln}x+\\text{ln}y[\/latex] to deduce that [latex]E(a+b)=E(a)E(b)[\/latex] for any real numbers [latex]a[\/latex], [latex]b[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572554258\" class=\"exercise\">\n<div id=\"fs-id1170572554260\" class=\"textbox\">\n<p id=\"fs-id1170572554263\">Pretend, for the moment, that we do not know that [latex]{e}^{x}[\/latex] is the inverse function of [latex]\\text{ln}x,[\/latex] but keep in mind that [latex]\\text{ln}x[\/latex] has an inverse function defined on [latex](\\text{\u2212}\\infty ,\\infty ).[\/latex] Call it <em>E<\/em>. Show that [latex]E\\text{'}(t)=E(t).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572309577\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572309577\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572309577\">[latex]x=E(\\text{ln}(x)).[\/latex] Then, [latex]1=\\frac{E\\text{'}(\\text{ln}x)}{x}\\text{or}x=E\\text{'}(\\text{ln}x).[\/latex] Since any number [latex]t[\/latex] can be written [latex]t=\\text{ln}x[\/latex] for some [latex]x[\/latex], and for such [latex]t[\/latex] we have [latex]x=E(t),[\/latex] it follows that for any [latex]t,E\\text{'}(t)=E(t).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1170571612025\" class=\"textbox\">\n<p id=\"fs-id1170571612028\">The sine integral, defined as [latex]S(x)={\\int }_{0}^{x}\\frac{ \\sin t}{t}dt[\/latex] is an important quantity in engineering. Although it does not have a simple closed formula, it is possible to estimate its behavior for large [latex]x[\/latex]. Show that for [latex]k\\ge 1,|S(2\\pi k)-S(2\\pi (k+1))|\\le \\frac{1}{k(2k+1)\\pi }.[\/latex] [latex](Hint\\text{:} \\sin (t+\\pi )=\\text{\u2212} \\sin t)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572373482\" class=\"exercise\">\n<div id=\"fs-id1170572373484\" class=\"textbox\">\n<p id=\"fs-id1170572373486\"><strong>[T]<\/strong> The normal distribution in probability is given by [latex]p(x)=\\frac{1}{\\sigma \\sqrt{2\\pi }}{e}^{\\text{\u2212}{(x-\\mu )}^{2}\\text{\/}2{\\sigma }^{2}},[\/latex] where <em>\u03c3<\/em> is the standard deviation and <em>\u03bc<\/em> is the average. The <em>standard normal distribution<\/em> in probability, [latex]{p}_{s},[\/latex] corresponds to [latex]\\mu =0\\text{ and }\\sigma =1.[\/latex] Compute the left endpoint estimates [latex]{R}_{10}\\text{ and }{R}_{100}[\/latex] of [latex]{\\int }_{-1}^{1}\\frac{1}{\\sqrt{2\\pi }}{e}^{\\text{\u2212}{x}^{2\\text{\/}2}}dx.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571652256\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571652256\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571652256\">[latex]{R}_{10}=0.6811,{R}_{100}=0.6827[\/latex]<\/p>\n<p><span id=\"fs-id1170571652285\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204259\/CNX_Calc_Figure_05_06_202.jpg\" alt=\"A graph of the function f(x) = .5 * ( sqrt(2)*e^(-.5x^2)) \/ sqrt(pi). It is a downward opening curve that is symmetric across the y axis, crossing at about (0, .4). It approaches 0 as x goes to positive and negative infinity. Between 1 and -1, ten rectangles are drawn for a right endpoint estimate of the area under the curve.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572380098\" class=\"exercise\">\n<div id=\"fs-id1170572380101\" class=\"textbox\">\n<p id=\"fs-id1170572380103\"><strong>[T]<\/strong> Compute the right endpoint estimates [latex]{R}_{50}\\text{ and }{R}_{100}[\/latex] of [latex]{\\int }_{-3}^{5}\\frac{1}{2\\sqrt{2\\pi }}{e}^{\\text{\u2212}{(x-1)}^{2}\\text{\/}8}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":311,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1769","chapter","type-chapter","status-publish","hentry"],"part":1684,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1769","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1769\/revisions"}],"predecessor-version":[{"id":2454,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1769\/revisions\/2454"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/parts\/1684"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1769\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/media?parent=1769"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapter-type?post=1769"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/contributor?post=1769"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/license?post=1769"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}