{"id":2031,"date":"2018-01-11T21:14:15","date_gmt":"2018-01-11T21:14:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/antiderivatives\/"},"modified":"2019-03-08T17:53:03","modified_gmt":"2019-03-08T17:53:03","slug":"antiderivatives","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/antiderivatives\/","title":{"raw":"4.10 Antiderivatives","rendered":"4.10 Antiderivatives"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Find the general antiderivative of a given function.<\/li>\r\n \t<li>Explain the terms and notation used for an indefinite integral.<\/li>\r\n \t<li>State the power rule for integrals.<\/li>\r\n \t<li>Use antidifferentiation to solve simple initial-value problems.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165042951286\">At this point, we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: Given a function [latex]f[\/latex], how do we find a function with the derivative [latex]f[\/latex] and why would we be interested in such a function?<\/p>\r\n<p id=\"fs-id1165043307884\">We answer the first part of this question by defining antiderivatives. The <strong>antiderivative<\/strong> of a function [latex]f[\/latex] is a function with a derivative [latex]f[\/latex]. Why are we interested in antiderivatives? The need for antiderivatives arises in many situations, and we look at various examples throughout the remainder of the text. Here we examine one specific example that involves rectilinear motion. In our examination in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction-4\/\">Derivatives<\/a> of rectilinear motion, we showed that given a position function [latex]s(t)[\/latex] of an object, then its velocity function [latex]v(t)[\/latex] is the derivative of [latex]s(t)[\/latex]\u2014that is, [latex]v(t)=s^{\\prime}(t)[\/latex]. Furthermore, the acceleration [latex]a(t)[\/latex] is the derivative of the velocity [latex]v(t)[\/latex]\u2014that is, [latex]a(t)=v^{\\prime}(t)=s^{\\prime \\prime}(t)[\/latex]. Now suppose we are given an acceleration function [latex]a[\/latex], but not the velocity function [latex]v[\/latex] or the position function [latex]s[\/latex]. Since [latex]a(t)=v^{\\prime}(t)[\/latex], determining the velocity function requires us to find an antiderivative of the acceleration function. Then, since [latex]v(t)=s^{\\prime}(t)[\/latex], determining the position function requires us to find an antiderivative of the velocity function. Rectilinear motion is just one case in which the need for antiderivatives arises. We will see many more examples throughout the remainder of the text. For now, let\u2019s look at the terminology and notation for antiderivatives, and determine the antiderivatives for several types of functions. We examine various techniques for finding antiderivatives of more complicated functions in the second volume of this text (<a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/HTmjSAcf@2.46:Z4WWhBaa@3\/Introduction\">Introduction to Techniques of Integration<\/a>).<\/p>\r\n\r\n<div id=\"fs-id1165043404679\" class=\"bc-section section\">\r\n<h1>The Reverse of Differentiation<\/h1>\r\n<p id=\"fs-id1165043323795\">At this point, we know how to find derivatives of various functions. We now ask the opposite question. Given a function [latex]f[\/latex], how can we find a function with derivative [latex]f[\/latex]? If we can find a function [latex]F[\/latex] with derivative [latex]f[\/latex], we call [latex]F[\/latex] an antiderivative of [latex]f[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165042478105\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1165042887564\">A function [latex]F[\/latex] is an antiderivative of the function [latex]f[\/latex] if<\/p>\r\n\r\n<div id=\"fs-id1165042945928\" class=\"equation unnumbered\">[latex]F^{\\prime}(x)=f(x)[\/latex]<\/div>\r\n<p id=\"fs-id1165042964886\">for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\r\n\r\n<\/div>\r\nConsider the function [latex]f(x)=2x[\/latex]. Knowing the power rule of differentiation, we conclude that [latex]F(x)=x^2[\/latex] is an antiderivative of [latex]f[\/latex] since [latex]F^{\\prime}(x)=2x[\/latex]. Are there any other antiderivatives of [latex]f[\/latex]? Yes; since the derivative of any constant [latex]C[\/latex] is zero, [latex]x^2+C[\/latex] is also an antiderivative of [latex]2x[\/latex]. Therefore, [latex]x^2+5[\/latex] and [latex]x^{2}-\\sqrt{2}[\/latex] are also antiderivatives. Are there any others that are not of the form [latex]x^2+C[\/latex] for some constant [latex]C[\/latex]? The answer is no. From Corollary 2 of the Mean Value Theorem, we know that if [latex]F[\/latex] and [latex]G[\/latex] are differentiable functions such that [latex]F^{\\prime}(x)=G^{\\prime}(x)[\/latex], then [latex]F(x)-G(x)=C[\/latex] for some constant [latex]C[\/latex]. This fact leads to the following important theorem.\r\n<div class=\"textbox key-takeaways theorem\">\r\n<h3>General Form of an Antiderivative<\/h3>\r\n<p id=\"fs-id1165042884523\">Let [latex]F[\/latex] be an antiderivative of [latex]f[\/latex] over an interval [latex]I[\/latex]. Then,<\/p>\r\n\r\n<ol id=\"fs-id1165043009392\">\r\n \t<li>for each constant [latex]C[\/latex], the function [latex]F(x)+C[\/latex] is also an antiderivative of [latex]f[\/latex] over [latex]I[\/latex];<\/li>\r\n \t<li>if [latex]G[\/latex] is an antiderivative of [latex]f[\/latex] over [latex]I[\/latex], there is a constant [latex]C[\/latex] for which [latex]G(x)=F(x)+C[\/latex] over [latex]I[\/latex].<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165042987370\">In other words, the most general form of the antiderivative of [latex]f[\/latex] over [latex]I[\/latex] is [latex]F(x)+C[\/latex].<\/p>\r\n\r\n<\/div>\r\nWe use this fact and our knowledge of derivatives to find all the antiderivatives for several functions.\r\n<div id=\"fs-id1165042616955\" class=\"textbox examples\">\r\n<h3>Finding Antiderivatives<\/h3>\r\n<div id=\"fs-id1165042984701\" class=\"exercise\">\r\n<div id=\"fs-id1165043431382\" class=\"textbox\">\r\n\r\nFor each of the following functions, find all antiderivatives.\r\n<ol id=\"fs-id1165043115403\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]f(x)=3x^2[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\frac{1}{x}[\/latex]<\/li>\r\n \t<li>[latex]f(x)= \\cos x[\/latex]<\/li>\r\n \t<li>[latex]f(x)=e^x[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"solution\">\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"46129\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"46129\"]a. Because\r\n\r\n[latex]\\frac{d}{dx}(x^3)=3x^2[\/latex]\r\n\r\nthen [latex]F(x)=x^3[\/latex] is an antiderivative of [latex]3x^2[\/latex]. Therefore, every antiderivative of [latex]3x^2[\/latex] is of the form [latex]x^3+C[\/latex] for some constant [latex]C[\/latex], and every function of the form [latex]x^3+C[\/latex] is an antiderivative of [latex]3x^2[\/latex].\r\n\r\nb. Let [latex]f(x)=\\ln |x|[\/latex]. For [latex]x&gt;0, \\, f(x)=\\ln (x)[\/latex] and\r\n\r\n[latex]\\frac{d}{dx}(\\ln x)=\\frac{1}{x}[\/latex].\r\n\r\nFor [latex]x&lt;0, \\, f(x)=\\ln (\u2212x)[\/latex] and\r\n\r\n[latex]\\frac{d}{dx}(\\ln (\u2212x))=-\\frac{1}{\u2212x}=\\frac{1}{x}[\/latex].\r\n\r\nTherefore,\r\n\r\n[latex]\\frac{d}{dx}(\\ln |x|)=\\frac{1}{x}[\/latex].\r\n\r\nThus, [latex]F(x)=\\ln |x|[\/latex] is an antiderivative of [latex]\\frac{1}{x}[\/latex]. Therefore, every antiderivative of [latex]\\frac{1}{x}[\/latex] is of the form [latex]\\ln |x|+C[\/latex] for some constant [latex]C[\/latex] and every function of the form [latex]\\ln |x|+C[\/latex] is an antiderivative of [latex]\\frac{1}{x}[\/latex].\r\n\r\nc. We have\r\n\r\n[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex],\r\n\r\nso [latex]F(x)= \\sin x[\/latex] is an antiderivative of [latex] \\cos x[\/latex]. Therefore, every antiderivative of [latex] \\cos x[\/latex] is of the form [latex] \\sin x+C[\/latex] for some constant [latex]C[\/latex] and every function of the form [latex] \\sin x+C[\/latex] is an antiderivative of [latex] \\cos x[\/latex].\r\n\r\nd. Since\r\n\r\n[latex]\\frac{d}{dx}(e^x)=e^x[\/latex],\r\n\r\nthen [latex]F(x)=e^x[\/latex] is an antiderivative of [latex]e^x[\/latex]. Therefore, every antiderivative of [latex]e^x[\/latex] is of the form [latex]e^x+C[\/latex] for some constant [latex]C[\/latex] and every function of the form [latex]e^x+C[\/latex] is an antiderivative of [latex]e^x[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043353933\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042966615\" class=\"exercise\">\r\n<div id=\"fs-id1165043194525\" class=\"textbox\">\r\n<p id=\"fs-id1165043353234\">Find all antiderivatives of [latex]f(x)= \\sin x[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<div class=\"textbox shaded\">\r\n<div class=\"solution\">\r\n<p id=\"fs-id1165042373760\">[reveal-answer q=\"314667\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"314667\"][latex]\u2212\\cos x+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043379984\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\nWhat function has a derivative of [latex] \\sin x[\/latex]?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042332032\" class=\"bc-section section\">\r\n<h1>Indefinite Integrals<\/h1>\r\n<p id=\"fs-id1165043344704\">We now look at the formal notation used to represent antiderivatives and examine some of their properties. These properties allow us to find antiderivatives of more complicated functions. Given a function [latex]f[\/latex], we use the notation [latex]f^{\\prime}(x)[\/latex] or [latex]\\frac{df}{dx}[\/latex] to denote the derivative of [latex]f[\/latex]. Here we introduce notation for antiderivatives. If [latex]F[\/latex] is an antiderivative of [latex]f[\/latex], we say that [latex]F(x)+C[\/latex] is the most general antiderivative of [latex]f[\/latex] and write<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\int f(x) dx=F(x)+C[\/latex].<\/div>\r\n<p id=\"fs-id1165042959838\">The symbol [latex]\\int [\/latex] is called an <em>integral sign<\/em>, and [latex]\\int f(x) dx[\/latex] is called the<strong> indefinite integral<\/strong> of [latex]f[\/latex].<\/p>\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1165043393369\">Given a function [latex]f[\/latex], the indefinite integral of [latex]f[\/latex], denoted<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\int f(x) dx[\/latex],<\/div>\r\nis the most general antiderivative of [latex]f[\/latex]. If [latex]F[\/latex] is an antiderivative of [latex]f[\/latex], then\r\n<div id=\"fs-id1165043119692\" class=\"equation unnumbered\">[latex]\\int f(x) dx=F(x)+C[\/latex].<\/div>\r\n<p id=\"fs-id1165043096049\">The expression [latex]f(x)[\/latex] is called the <em>integrand<\/em> and the variable [latex]x[\/latex] is the <em>variable of integration<\/em>.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1165042558766\">Given the terminology introduced in this definition, the act of finding the antiderivatives of a function [latex]f[\/latex] is usually referred to as <em>integrating<\/em> [latex]f[\/latex].<\/p>\r\n<p id=\"fs-id1165042936505\">For a function [latex]f[\/latex] and an antiderivative [latex]F[\/latex], the functions [latex]F(x)+C[\/latex], where [latex]C[\/latex] is any real number, is often referred to as <em>the family of antiderivatives of<\/em> [latex]f[\/latex]. For example, since [latex]x^2[\/latex] is an antiderivative of [latex]2x[\/latex] and any antiderivative of [latex]2x[\/latex] is of the form [latex]x^2+C[\/latex], we write<\/p>\r\n\r\n<div id=\"fs-id1165043327744\" class=\"equation unnumbered\">[latex]\\int 2x dx=x^2+C[\/latex].<\/div>\r\n<p id=\"fs-id1165043097500\">The collection of all functions of the form [latex]x^2+C[\/latex], where [latex]C[\/latex] is any real number, is known as the <em>family of antiderivatives of<\/em> [latex]2x[\/latex].\u00a0<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_10_001\">(Figure)<\/a> shows a graph of this family of antiderivatives.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_04_10_001\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"646\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211358\/CNX_Calc_Figure_04_10_001.jpg\" alt=\"The graphs for y = x2 + 2, y = x2 + 1, y = x2, y = x2 \u2212 1, and y = x2 \u2212 2 are shown.\" width=\"646\" height=\"575\" \/> <strong>Figure 1.<\/strong> The family of antiderivatives of [latex]2x[\/latex] consists of all functions of the form [latex]x^2+C[\/latex], where [latex]C[\/latex] is any real number.[\/caption]<\/div>\r\nFor some functions, evaluating indefinite integrals follows directly from properties of derivatives. For example, for [latex]n \\ne \u22121[\/latex],\r\n<div id=\"fs-id1165043350427\" class=\"equation unnumbered\">[latex]\\int x^n dx=\\frac{x^{n+1}}{n+1}+C[\/latex],<\/div>\r\n<p id=\"fs-id1165043015098\">which comes directly from<\/p>\r\n\r\n<div id=\"fs-id1165043036022\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}(\\frac{x^{n+1}}{n+1})=(n+1)\\frac{x^n}{n+1}=x^n[\/latex].<\/div>\r\n<p id=\"fs-id1165042343281\">This fact is known as <em>the power rule for integrals<\/em>.<\/p>\r\n\r\n<div id=\"fs-id1165043041347\" class=\"textbox key-takeaways theorem\">\r\n<h3>Power Rule for Integrals<\/h3>\r\n<p id=\"fs-id1165042514785\">For [latex]n \\ne \u22121[\/latex],<\/p>\r\n\r\n<div id=\"fs-id1165043250161\" class=\"equation unnumbered\">[latex]\\int x^n dx=\\frac{x^{n+1}}{n+1}+C[\/latex].<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165043385541\">Evaluating indefinite integrals for some other functions is also a straightforward calculation. The following table lists the indefinite integrals for several common functions. A more complete list appears in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/back-matter\/table-of-derivatives\/\">Appendix B<\/a>.<\/p>\r\n\r\n<table summary=\"This is a table with two columns and fourteen rows, titled \u201cIntegration Formulas.\u201d The first row is a header row, and labels column one \u201cDifferentiation Formula\u201d and column two \u201cIndefinite Integral.\u201d The second row reads d\/dx (k) = 0, the integral of kdx = the integral of kx^0dx = kx + C. The third row reads d\/dx(x^n) = nx^(x-1), the integral of x^ndn = (x^n+1)\/(n+1) + C for n is not equal to negative 1. The fourth row reads d\/dx(ln(the absolute value of x))=1\/x, the integral of (1\/x)dx = ln(the absolute value of x) + C. The fifth row reads d\/dx(e^x) = e^x, the integral of e^xdx = e^x + C. The sixth row reads d\/dx(sinx) = cosx, the integral of cosxdx = sinx + C. The seventh row reads d\/dx(cosx) = negative sinx, the integral of sinxdx = negative cosx + C. The eighth row reads d\/dx(tanx) = sec squared x, the integral of sec squared xdx = tanx + C. The ninth row reads d\/dx(cscx) = negative cscxcotx, the integral of cscxcotxdx = negative cscx + C. The tenth row reads d\/dx(secx) = secxtanx, the integral of secxtanxdx = secx + C. The eleventh row reads d\/dx(cotx) = negative csc squared x, the integral of csc squared xdx = negative cot x + C. The twelfth row reads d\/dx(sin^-1(x)) = 1\/the square root of (1 \u2013 x^2), the integral of 1\/(the square root of (x^2 \u2013 1) = sin^-1(x) + C. The thirteenth row reads d\/dx (tan^-1(x)) = 1\/(1 + x^2), the integral of 1\/(1 + x^2)dx = tan^-1(x) + C. The fourteenth row reads d\/dx(sec^-1(the absolute value of x)) = 1\/x(the square root of x^2 \u2013 1), the integral of 1\/x(the square root of x^2 \u2013 1)dx = sec^-1(the absolute value of x) + C.\"><caption>Integration Formulas<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Differentiation Formula<\/th>\r\n<th>Indefinite Integral<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(k)=0[\/latex]<\/td>\r\n<td>[latex]\\int kdx=\\int kx^0 dx=kx+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(x^n)=nx^{n-1}[\/latex]<\/td>\r\n<td>[latex]\\int x^n dx=\\frac{x^{n+1}}{n+1}+C[\/latex] for [latex]n\\ne \u22121[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\ln |x|)=\\frac{1}{x}[\/latex]<\/td>\r\n<td>[latex]\\int \\frac{1}{x}dx=\\ln |x|+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(e^x)=e^x[\/latex]<\/td>\r\n<td>[latex]\\int e^x dx=e^x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/td>\r\n<td>[latex]\\int \\cos x dx= \\sin x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\cos x)=\u2212 \\sin x[\/latex]<\/td>\r\n<td>[latex]\\int \\sin x dx=\u2212 \\cos x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\tan x)= \\sec^2 x[\/latex]<\/td>\r\n<td>[latex]\\int \\sec^2 x dx= \\tan x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/td>\r\n<td>[latex]\\int \\csc x \\cot x dx=\u2212\\csc x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/td>\r\n<td>[latex]\\int \\sec x \\tan x dx= \\sec x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/td>\r\n<td>[latex]\\int \\csc^2 x dx=\u2212\\cot x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}( \\sin^{-1} x)=\\frac{1}{\\sqrt{1-x^2}}[\/latex]<\/td>\r\n<td>[latex]\\int \\frac{1}{\\sqrt{1-x^2}} dx= \\sin^{-1} x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\tan^{-1} x)=\\frac{1}{1+x^2}[\/latex]<\/td>\r\n<td>[latex]\\int \\frac{1}{1+x^2} dx= \\tan^{-1} x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\sec^{-1} |x|)=\\frac{1}{x\\sqrt{x^2-1}}[\/latex]<\/td>\r\n<td>[latex]\\int \\frac{1}{x\\sqrt{x^2-1}} dx= \\sec^{-1} |x|+C[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042328678\">From the definition of indefinite integral of [latex]f[\/latex], we know<\/p>\r\n\r\n<div id=\"fs-id1165042373302\" class=\"equation unnumbered\">[latex]\\int f(x) dx=F(x)+C[\/latex]<\/div>\r\n<p id=\"fs-id1165042322393\">if and only if [latex]F[\/latex] is an antiderivative of [latex]f[\/latex]. Therefore, when claiming that<\/p>\r\n\r\n<div id=\"fs-id1165042472053\" class=\"equation unnumbered\">[latex]\\int f(x) dx=F(x)+C[\/latex]<\/div>\r\n<p id=\"fs-id1165042375807\">it is important to check whether this statement is correct by verifying that [latex]F^{\\prime}(x)=f(x)[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165043393824\" class=\"textbox examples\">\r\n<h3>Verifying an Indefinite Integral<\/h3>\r\n<div id=\"fs-id1165043393826\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1165043428255\">Each of the following statements is of the form [latex]\\int f(x) dx=F(x)+C[\/latex]. Verify that each statement is correct by showing that [latex]F^{\\prime}(x)=f(x)[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1165043281729\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\int (x+e^x) dx=\\frac{x^2}{2}+e^x+C[\/latex]<\/li>\r\n \t<li>[latex]\\int xe^xdx=xe^x-e^x+C[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042710847\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042710847\"]\r\n<ol id=\"fs-id1165042710847\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Since\r\n<div id=\"fs-id1165042710856\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}(\\frac{x^2}{2}+e^x+C)=x+e^x[\/latex],<\/div>\r\nthe statement\r\n<div id=\"fs-id1165042319135\" class=\"equation unnumbered\">[latex]\\int (x+e^x)dx=\\frac{x^2}{2}+e^x+C[\/latex]<\/div>\r\nis correct.\r\nNote that we are verifying an indefinite integral for a sum. Furthermore, [latex]\\frac{x^2}{2}[\/latex] and [latex]e^x[\/latex] are antiderivatives of [latex]x[\/latex] and [latex]e^x[\/latex], respectively, and the sum of the antiderivatives is an antiderivative of the sum. We discuss this fact again later in this section.<\/li>\r\n \t<li>Using the product rule, we see that\r\n<div class=\"equation unnumbered\">[latex]\\frac{d}{dx}(xe^x-e^x+C)=e^x+xe^x-e^x=xe^x[\/latex].<\/div>\r\nTherefore, the statement\r\n<div id=\"fs-id1165043257181\" class=\"equation unnumbered\">[latex]\\int xe^x dx=xe^x-e^x+C[\/latex]<\/div>\r\nis correct.\r\nNote that we are verifying an indefinite integral for a product. The antiderivative [latex]xe^x-e^x[\/latex] is not a product of the antiderivatives. Furthermore, the product of antiderivatives, [latex]x^2 e^x\/2[\/latex] is not an antiderivative of [latex]xe^x[\/latex] since\r\n<div id=\"fs-id1165042320874\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}(\\frac{x^2e^x}{2})=xe^x+\\frac{x^2e^x}{2} \\ne xe^x[\/latex].<\/div>\r\nIn general, the product of antiderivatives is not an antiderivative of a product.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043078178\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165043078181\" class=\"exercise\">\r\n<div id=\"fs-id1165042364598\" class=\"textbox\">\r\n<p id=\"fs-id1165042364600\">Verify that [latex]\\int x \\cos x dx=x \\sin x+ \\cos x+C[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043257533\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043257533\"]\r\n<p id=\"fs-id1165043257533\">[latex]\\frac{d}{dx}(x \\sin x+ \\cos x+C)= \\sin x+x \\cos x- \\sin x=x \\cos x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043219874\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165043219881\">Calculate [latex]\\frac{d}{dx}(x \\sin x+ \\cos x+C)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042318564\">In <a class=\"autogenerated-content\" href=\"#fs-id1165043092431\">(Figure)<\/a>, we listed the indefinite integrals for many elementary functions. Let\u2019s now turn our attention to evaluating indefinite integrals for more complicated functions. For example, consider finding an antiderivative of a sum [latex]f+g[\/latex]. In <a class=\"autogenerated-content\" href=\"#fs-id1165043393824\">(Figure)<\/a>a. we showed that an antiderivative of the sum [latex]x+e^x[\/latex] is given by the sum [latex](\\frac{x^2}{2})+e^x[\/latex]\u2014that is, an antiderivative of a sum is given by a sum of antiderivatives. This result was not specific to this example. In general, if [latex]F[\/latex] and [latex]G[\/latex] are antiderivatives of any functions [latex]f[\/latex] and [latex]g[\/latex], respectively, then<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\frac{d}{dx}(F(x)+G(x))=F^{\\prime}(x)+G^{\\prime}(x)=f(x)+g(x)[\/latex].<\/div>\r\n<p id=\"fs-id1165043372671\">Therefore, [latex]F(x)+G(x)[\/latex] is an antiderivative of [latex]f(x)+g(x)[\/latex] and we have<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\int (f(x)+g(x)) dx=F(x)+G(x)+C[\/latex].<\/div>\r\n<p id=\"fs-id1165043174082\">Similarly,<\/p>\r\n\r\n<div id=\"fs-id1165043174085\" class=\"equation unnumbered\">[latex]\\int (f(x)-g(x)) dx=F(x)-G(x)+C[\/latex].<\/div>\r\n<p id=\"fs-id1165042383134\">In addition, consider the task of finding an antiderivative of [latex]kf(x)[\/latex], where [latex]k[\/latex] is any real number. Since<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\frac{d}{dx}(kf(x))=k\\frac{d}{dx}F(x)=kF^{\\prime}(x)[\/latex]<\/div>\r\n<p id=\"fs-id1165043222034\">for any real number [latex]k[\/latex], we conclude that<\/p>\r\n\r\n<div id=\"fs-id1165042383898\" class=\"equation unnumbered\">[latex]\\int kf(x) dx=kF(x)+C[\/latex].<\/div>\r\n<p id=\"fs-id1165043425482\">These properties are summarized next.<\/p>\r\n\r\n<div id=\"fs-id1165043425485\" class=\"textbox key-takeaways theorem\">\r\n<h3>Properties of Indefinite Integrals<\/h3>\r\n<p id=\"fs-id1165043395041\">Let [latex]F[\/latex] and [latex]G[\/latex] be antiderivatives of [latex]f[\/latex] and [latex]g[\/latex], respectively, and let [latex]k[\/latex] be any real number.<\/p>\r\n<p id=\"fs-id1165043393659\">Sums and Differences<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\int (f(x) \\pm g(x)) dx=F(x) \\pm G(x)+C[\/latex]<\/div>\r\n<p id=\"fs-id1165042328714\">Constant Multiples<\/p>\r\n\r\n<div id=\"fs-id1165042328717\" class=\"equation unnumbered\">[latex]\\int kf(x) dx=kF(x)+C[\/latex]<\/div>\r\n<\/div>\r\nFrom this theorem, we can evaluate any integral involving a sum, difference, or constant multiple of functions with antiderivatives that are known. Evaluating integrals involving products, quotients, or compositions is more complicated (see <a class=\"autogenerated-content\" href=\"#fs-id1165043393824\">(Figure)<\/a>b. for an example involving an antiderivative of a product.) We look at and address integrals involving these more complicated functions in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction-3\/\">Introduction to Integration<\/a>. In the next example, we examine how to use this theorem to calculate the indefinite integrals of several functions.\r\n<div id=\"fs-id1165043248811\" class=\"textbox examples\">\r\n<h3>Evaluating Indefinite Integrals<\/h3>\r\n<div id=\"fs-id1165042705917\" class=\"exercise\">\r\n<div id=\"fs-id1165042705919\" class=\"textbox\">\r\n\r\nEvaluate each of the following indefinite integrals:\r\n<ol style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\int (5x^3-7x^2+3x+4) dx[\/latex]<\/li>\r\n \t<li>[latex]\\int \\frac{x^2+4\\sqrt[3]{x}}{x} dx[\/latex]<\/li>\r\n \t<li>[latex]\\int \\frac{4}{1+x^2} dx[\/latex]<\/li>\r\n \t<li>[latex]\\int \\tan x \\cos x dx[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042552215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042552215\"]\r\n<ol id=\"fs-id1165042552215\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Using <a class=\"autogenerated-content\" href=\"#fs-id1165043425485\">(Figure)<\/a>, we can integrate each of the four terms in the integrand separately. We obtain\r\n<div id=\"fs-id1165042552227\" class=\"equation unnumbered\">[latex]\\int (5x^3-7x^2+3x+4) dx=\\int 5x^3 dx-\\int 7x^2 dx+\\int 3x dx+\\int 4 dx[\/latex].<\/div>\r\nFrom the second part of <a class=\"autogenerated-content\" href=\"#fs-id1165043425485\">(Figure)<\/a>, each coefficient can be written in front of the integral sign, which gives\r\n<div id=\"fs-id1165043312575\" class=\"equation unnumbered\">[latex]\\int 5x^3 dx-\\int 7x^2 dx+\\int 3x dx+\\int 4 dx=5\\int x^3 dx-7\\int x^2 dx+3\\int x dx+4\\int 1 dx[\/latex].<\/div>\r\nUsing the power rule for integrals, we conclude that\r\n<div id=\"fs-id1165042407363\" class=\"equation unnumbered\">[latex]\\int (5x^3-7x^2+3x+4) dx=\\frac{5}{4}x^4-\\frac{7}{3}x^3+\\frac{3}{2}x^2+4x+C[\/latex].<\/div><\/li>\r\n \t<li>Rewrite the integrand as\r\n<div id=\"fs-id1165042371846\" class=\"equation unnumbered\">[latex]\\frac{x^2+4\\sqrt[3]{x}}{x}=\\frac{x^2}{x}+\\frac{4\\sqrt[3]{x}}{x}[\/latex].<\/div>\r\nThen, to evaluate the integral, integrate each of these terms separately. Using the power rule, we have\r\n<div id=\"fs-id1165043427498\" class=\"equation unnumbered\">[latex]\\begin{array}{ll} \\int (x+\\frac{4}{x^{2\/3}}) dx &amp; =\\int x dx+4\\int x^{-2\/3} dx \\\\ &amp; =\\frac{1}{2}x^2+4\\frac{1}{(\\frac{-2}{3})+1}x^{(-2\/3)+1}+C \\\\ &amp; =\\frac{1}{2}x^2+12x^{1\/3}+C \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Using <a class=\"autogenerated-content\" href=\"#fs-id1165043425485\">(Figure)<\/a>, write the integral as\r\n<div id=\"fs-id1165043348665\" class=\"equation unnumbered\">[latex]4\\int \\frac{1}{1+x^2} dx[\/latex].<\/div>\r\nThen, use the fact that [latex] \\tan^{-1} (x)[\/latex] is an antiderivative of [latex]\\frac{1}{1+x^2}[\/latex] to conclude that\r\n<div id=\"fs-id1165042374764\" class=\"equation unnumbered\">[latex]\\int \\frac{4}{1+x^2} dx=4 \\tan^{-1} (x)+C[\/latex].<\/div><\/li>\r\n \t<li>Rewrite the integrand as\r\n<div class=\"equation unnumbered\">[latex] \\tan x \\cos x=\\frac{ \\sin x}{ \\cos x} \\cos x= \\sin x[\/latex].<\/div>\r\nTherefore,\r\n<div id=\"fs-id1165043317182\" class=\"equation unnumbered\">[latex]\\int \\tan x \\cos x dx=\\int \\sin x dx=\u2212 \\cos x+C[\/latex].<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165043426269\" class=\"exercise\">\r\n<div id=\"fs-id1165043426271\" class=\"textbox\">\r\n\r\nEvaluate [latex]\\int (4x^3-5x^2+x-7) dx[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043259694\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043259694\"]\r\n<p id=\"fs-id1165043259694\">[latex]x^4-\\frac{5}{3}x^3+\\frac{1}{2}x^2-7x+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042468208\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042323621\">Integrate each term in the integrand separately, making use of the power rule.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042323629\" class=\"bc-section section\">\r\n<h1>Initial-Value Problems<\/h1>\r\n<p id=\"fs-id1165042323634\">We look at techniques for integrating a large variety of functions involving products, quotients, and compositions later in the text. Here we turn to one common use for antiderivatives that arises often in many applications: solving differential equations.<\/p>\r\n<p id=\"fs-id1165042323639\">A <em>differential equation<\/em> is an equation that relates an unknown function and one or more of its derivatives. The equation<\/p>\r\n\r\n<div id=\"fs-id1165042349924\" class=\"equation\">[latex]\\frac{dy}{dx}=f(x)[\/latex]<\/div>\r\n<p id=\"fs-id1165042349952\">is a simple example of a differential equation. Solving this equation means finding a function [latex]y[\/latex] with a derivative [latex]f[\/latex]. Therefore, the solutions of <a class=\"autogenerated-content\" href=\"#fs-id1165042349924\">(Figure)<\/a> are the antiderivatives of [latex]f[\/latex]. If [latex]F[\/latex] is one antiderivative of [latex]f[\/latex], every function of the form [latex]y=F(x)+C[\/latex] is a solution of that differential equation. For example, the solutions of<\/p>\r\n\r\n<div id=\"fs-id1165042323537\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=6x^2[\/latex]<\/div>\r\nare given by\r\n<div id=\"fs-id1165043431000\" class=\"equation unnumbered\">[latex]y=\\int 6x^2 dx=2x^3+C[\/latex].<\/div>\r\n<p id=\"fs-id1165042407328\">Sometimes we are interested in determining whether a particular solution curve passes through a certain point [latex](x_0,y_0)[\/latex]\u2014that is, [latex]y(x_0)=y_0[\/latex]. The problem of finding a function [latex]y[\/latex] that satisfies a differential equation<\/p>\r\n\r\n<div id=\"fs-id1165042375659\" class=\"equation\">[latex]\\frac{dy}{dx}=f(x)[\/latex]<\/div>\r\n<p id=\"fs-id1165042368468\">with the additional condition<\/p>\r\n\r\n<div id=\"fs-id1165042368471\" class=\"equation\">[latex]y(x_0)=y_0[\/latex]<\/div>\r\nis an example of an <strong>initial-value problem<\/strong>. The condition [latex]y(x_0)=y_0[\/latex] is known as an <em>initial condition<\/em>. For example, looking for a function [latex]y[\/latex] that satisfies the differential equation\r\n<div id=\"fs-id1165042545822\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=6x^2[\/latex]<\/div>\r\n<p id=\"fs-id1165043393696\">and the initial condition<\/p>\r\n\r\n<div id=\"fs-id1165043393699\" class=\"equation unnumbered\">[latex]y(1)=5[\/latex]<\/div>\r\n<p id=\"fs-id1165043393719\">is an example of an initial-value problem. Since the solutions of the differential equation are [latex]y=2x^3+C[\/latex], to find a function [latex]y[\/latex] that also satisfies the initial condition, we need to find [latex]C[\/latex] such that [latex]y(1)=2(1)^3+C=5[\/latex]. From this equation, we see that [latex]C=3[\/latex], and we conclude that [latex]y=2x^3+3[\/latex] is the solution of this initial-value problem as shown in the following graph.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_04_10_002\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"641\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211402\/CNX_Calc_Figure_04_10_002.jpg\" alt=\"The graphs for y = 2x3 + 6, y = 2x3 + 3, y = 2x3, and y = 2x3 \u2212 3 are shown.\" width=\"641\" height=\"497\" \/> <strong>Figure 2.<\/strong> Some of the solution curves of the differential equation [latex]\\frac{dy}{dx}=6x^2[\/latex] are displayed. The function [latex]y=2x^3+3[\/latex] satisfies the differential equation and the initial condition [latex]y(1)=5[\/latex].[\/caption]<\/div>\r\n<div id=\"fs-id1165043327648\" class=\"textbox examples\">\r\n<h3>Solving an Initial-Value Problem<\/h3>\r\n<div id=\"fs-id1165043327650\" class=\"exercise\">\r\n<div id=\"fs-id1165043327652\" class=\"textbox\">\r\n<p id=\"fs-id1165043327658\">Solve the initial-value problem<\/p>\r\n\r\n<div id=\"fs-id1165043327661\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}= \\sin x, \\, y(0)=5[\/latex].<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043430929\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043430929\"]\r\n<p id=\"fs-id1165043430929\">First we need to solve the differential equation. If [latex]\\frac{dy}{dx}= \\sin x,[\/latex] then<\/p>\r\n\r\n<div id=\"fs-id1165043286683\" class=\"equation unnumbered\">[latex]y=\\int \\sin (x) dx=\u2212 \\cos x+C[\/latex].<\/div>\r\n<p id=\"fs-id1165042318817\">Next we need to look for a solution [latex]y[\/latex] that satisfies the initial condition. The initial condition [latex]y(0)=5[\/latex] means we need a constant [latex]C[\/latex] such that [latex]\u2212 \\cos x+C=5[\/latex]. Therefore,<\/p>\r\n\r\n<div id=\"fs-id1165043424804\" class=\"equation unnumbered\">[latex]C=5+ \\cos (0)=6[\/latex].<\/div>\r\n<p id=\"fs-id1165043424835\">The solution of the initial-value problem is [latex]y=\u2212 \\cos x+6[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042327334\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042327337\" class=\"exercise\">\r\n<div id=\"fs-id1165042327339\" class=\"textbox\">\r\n<p id=\"fs-id1165042327341\">Solve the initial value problem [latex]\\frac{dy}{dx}=3x^{-2}, \\, y(1)=2[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043327482\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043327482\"]\r\n<p id=\"fs-id1165043327482\">[latex]y=-\\frac{3}{x}+5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042640833\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042640839\">Find all antiderivatives of [latex]f(x)=3x^{-2}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165043174043\">Initial-value problems arise in many applications. Next we consider a problem in which a driver applies the brakes in a car. We are interested in how long it takes for the car to stop. Recall that the velocity function [latex]v(t)[\/latex] is the derivative of a position function [latex]s(t)[\/latex], and the acceleration [latex]a(t)[\/latex] is the derivative of the velocity function. In earlier examples in the text, we could calculate the velocity from the position and then compute the acceleration from the velocity. In the next example we work the other way around. Given an acceleration function, we calculate the velocity function. We then use the velocity function to determine the position function.<\/p>\r\n\r\n<div id=\"fs-id1165042640762\" class=\"textbox examples\">\r\n<h3>Decelerating Car<\/h3>\r\n<div id=\"fs-id1165042640764\" class=\"exercise\">\r\n<div id=\"fs-id1165042640767\" class=\"textbox\">\r\n<p id=\"fs-id1165042640772\">A car is traveling at the rate of 88 ft\/sec (60 mph) when the brakes are applied. The car begins decelerating at a constant rate of 15 ft\/sec<sup>2<\/sup>.<\/p>\r\n\r\n<ol id=\"fs-id1165043430821\" style=\"list-style-type: lower-alpha\">\r\n \t<li>How many seconds elapse before the car stops?<\/li>\r\n \t<li>How far does the car travel during that time?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043430836\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043430836\"]\r\n<ol id=\"fs-id1165043430836\" style=\"list-style-type: lower-alpha\">\r\n \t<li>First we introduce variables for this problem. Let [latex]t[\/latex] be the time (in seconds) after the brakes are first applied. Let [latex]a(t)[\/latex] be the acceleration of the car (in feet per seconds squared) at time [latex]t[\/latex]. Let [latex]v(t)[\/latex] be the velocity of the car (in feet per second) at time [latex]t[\/latex]. Let [latex]s(t)[\/latex] be the car\u2019s position (in feet) beyond the point where the brakes are applied at time [latex]t[\/latex].\r\nThe car is traveling at a rate of 88 ft\/sec. Therefore, the initial velocity is [latex]v(0)=88[\/latex] ft\/sec. Since the car is decelerating, the acceleration is\r\n<div id=\"fs-id1165043323894\" class=\"equation unnumbered\">[latex]a(t)=-15[\/latex] ft\/sec<sup>2<\/sup><\/div>\r\nThe acceleration is the derivative of the velocity,\r\n<div id=\"fs-id1165043219161\" class=\"equation unnumbered\">[latex]v^{\\prime}(t)=-15[\/latex].<\/div>\r\nTherefore, we have an initial-value problem to solve:\r\n<div id=\"fs-id1165043219190\" class=\"equation unnumbered\">[latex]v^{\\prime}(t)=-15, \\, v(0)=88[\/latex].<\/div>\r\nIntegrating, we find that\r\n<div id=\"fs-id1165043395175\" class=\"equation unnumbered\">[latex]v(t)=-15t+C[\/latex].<\/div>\r\nSince [latex]v(0)=88, \\, C=88[\/latex]. Thus, the velocity function is\r\n<div id=\"fs-id1165042373710\" class=\"equation unnumbered\">[latex]v(t)=-15t+88[\/latex].<\/div>\r\nTo find how long it takes for the car to stop, we need to find the time [latex]t[\/latex] such that the velocity is zero. Solving [latex]-15t+88=0[\/latex], we obtain [latex]t=\\frac{88}{15}[\/latex] sec.<\/li>\r\n \t<li>To find how far the car travels during this time, we need to find the position of the car after [latex]\\frac{88}{15}[\/latex] sec. We know the velocity [latex]v(t)[\/latex] is the derivative of the position [latex]s(t)[\/latex]. Consider the initial position to be [latex]s(0)=0[\/latex]. Therefore, we need to solve the initial-value problem\r\n<div id=\"fs-id1165043380494\" class=\"equation unnumbered\">[latex]s^{\\prime}(t)=-15t+88, \\, s(0)=0[\/latex].<\/div>\r\nIntegrating, we have\r\n<div id=\"fs-id1165043317270\" class=\"equation unnumbered\">[latex]s(t)=-\\frac{15}{2}t^2+88t+C[\/latex].<\/div>\r\nSince [latex]s(0)=0[\/latex], the constant is [latex]C=0[\/latex]. Therefore, the position function is\r\n<div id=\"fs-id1165043250997\" class=\"equation unnumbered\">[latex]s(t)=-\\frac{15}{2}t^2+88t[\/latex].<\/div>\r\nAfter [latex]t=\\frac{88}{15}[\/latex] sec, the position is [latex]s(\\frac{88}{15})\\approx 258.133[\/latex] ft.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042708223\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042708226\" class=\"exercise\">\r\n<div id=\"fs-id1165042708229\" class=\"textbox\">\r\n<p id=\"fs-id1165042708231\">Suppose the car is traveling at the rate of 44 ft\/sec. How long does it take for the car to stop? How far will the car travel?<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<div class=\"textbox shaded\">\r\n<div class=\"solution\">\r\n<p id=\"fs-id1165042708242\">[reveal-answer q=\"923849\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"923849\"]2.93 \\sec, 64.5 ft[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<div class=\"commentary\"><\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165043396281\">[latex]v(t)=-15t+44[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043396312\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165042323569\">\r\n \t<li>If [latex]F[\/latex] is an antiderivative of [latex]f[\/latex], then every antiderivative of [latex]f[\/latex] is of the form [latex]F(x)+C[\/latex] for some constant [latex]C[\/latex].<\/li>\r\n \t<li>Solving the initial-value problem\r\n<div id=\"fs-id1165043259810\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=f(x),y(x_0)=y_0[\/latex]<\/div>\r\nrequires us first to find the set of antiderivatives of [latex]f[\/latex] and then to look for the particular antiderivative that also satisfies the initial condition.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165042709600\" class=\"textbox exercises\">\r\n<p id=\"fs-id1165042709603\">For the following exercises, show that [latex]F(x)[\/latex] is an antiderivative of [latex]f(x)[\/latex].<\/p>\r\n\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1165042333423\" class=\"textbox\">\r\n<p id=\"fs-id1165042333425\"><strong>1.\u00a0<\/strong>[latex]F(x)=5x^3+2x^2+3x+1, \\, f(x)=15x^2+4x+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042465569\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042465569\"]\r\n<p id=\"fs-id1165042465569\">[latex]F^{\\prime}(x)=15x^2+4x+3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042364150\" class=\"exercise\">\r\n<div id=\"fs-id1165042364152\" class=\"textbox\">\r\n<p id=\"fs-id1165042364154\"><strong>2.\u00a0<\/strong>[latex]F(x)=x^2+4x+1, \\, f(x)=2x+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043317346\" class=\"exercise\">\r\n<div id=\"fs-id1165043317348\" class=\"textbox\">\r\n<p id=\"fs-id1165043317350\"><strong>3.\u00a0<\/strong>[latex]F(x)=x^2e^x, \\, f(x)=e^x(x^2+2x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043327399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043327399\"]\r\n<p id=\"fs-id1165043327399\">[latex]F^{\\prime}(x)=2xe^x+x^2e^x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042659411\" class=\"exercise\">\r\n<div id=\"fs-id1165042659413\" class=\"textbox\">\r\n<p id=\"fs-id1165042659415\"><strong>4.\u00a0<\/strong>[latex]F(x)= \\cos x, \\, f(x)=\u2212 \\sin x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042640878\" class=\"exercise\">\r\n<div id=\"fs-id1165042640880\" class=\"textbox\">\r\n<p id=\"fs-id1165042640882\"><strong>5.\u00a0<\/strong>[latex]F(x)=e^x, \\, f(x)=e^x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042640926\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042640926\"]\r\n<p id=\"fs-id1165042640926\">[latex]F^{\\prime}(x)=e^x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042638498\">For the following exercises, find the antiderivative of the function.<\/p>\r\n\r\n<div id=\"fs-id1165042638501\" class=\"exercise\">\r\n<div id=\"fs-id1165042638503\" class=\"textbox\">\r\n<p id=\"fs-id1165042638505\"><strong>6.\u00a0<\/strong>[latex]f(x)=\\frac{1}{x^2}+x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043390814\" class=\"exercise\">\r\n<div id=\"fs-id1165043390816\" class=\"textbox\">\r\n<p id=\"fs-id1165043390818\"><strong>7.\u00a0<\/strong>[latex]f(x)=e^x-3x^2+ \\sin x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043390859\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043390859\"]\r\n<p id=\"fs-id1165043390859\">[latex]F(x)=e^x-x^3- \\cos (x)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042708318\" class=\"exercise\">\r\n<div id=\"fs-id1165042708321\" class=\"textbox\">\r\n\r\n<strong>8.\u00a0<\/strong>[latex]f(x)=e^x+3x-x^2[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1165042659509\" class=\"textbox\">\r\n<p id=\"fs-id1165042659512\"><strong>9.\u00a0<\/strong>[latex]f(x)=x-1+4 \\sin (2x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042659554\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042659554\"]\r\n<p id=\"fs-id1165042659554\">[latex]F(x)=\\frac{x^2}{2}-x-2 \\cos (2x)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042465656\">For the following exercises, find the antiderivative [latex]F(x)[\/latex] of each function [latex]f(x)[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165042465686\" class=\"exercise\">\r\n<div id=\"fs-id1165042465688\" class=\"textbox\">\r\n<p id=\"fs-id1165042465690\"><strong>10.\u00a0<\/strong>[latex]f(x)=5x^4+4x^5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043427405\" class=\"exercise\">\r\n<div id=\"fs-id1165043427407\" class=\"textbox\">\r\n<p id=\"fs-id1165043427409\"><strong>11.\u00a0<\/strong>[latex]f(x)=x+12x^2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043108238\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043108238\"]\r\n<p id=\"fs-id1165043108238\">[latex]F(x)=\\frac{1}{2}x^2+4x^3+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043108281\" class=\"exercise\">\r\n<div id=\"fs-id1165043108283\" class=\"textbox\">\r\n<p id=\"fs-id1165043108285\"><strong>12.\u00a0<\/strong>[latex]f(x)=\\frac{1}{\\sqrt{x}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042657750\" class=\"exercise\">\r\n<div id=\"fs-id1165042657752\" class=\"textbox\">\r\n<p id=\"fs-id1165042657754\"><strong>13.\u00a0<\/strong>[latex]f(x)=(\\sqrt{x})^3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042657787\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042657787\"]\r\n<p id=\"fs-id1165042657787\">[latex]F(x)=\\frac{2}{5}(\\sqrt{x})^5+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043219223\" class=\"exercise\">\r\n<div id=\"fs-id1165043219225\" class=\"textbox\">\r\n<p id=\"fs-id1165043219227\"><strong>14.\u00a0<\/strong>[latex]f(x)=x^{1\/3}+(2x)^{1\/3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042707199\" class=\"exercise\">\r\n<div id=\"fs-id1165042707201\" class=\"textbox\">\r\n<p id=\"fs-id1165042707203\"><strong>15.\u00a0<\/strong>[latex]f(x)=\\frac{x^{1\/3}}{x^{2\/3}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042707247\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042707247\"]\r\n<p id=\"fs-id1165042707247\">[latex]F(x)=\\frac{3}{2}x^{2\/3}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043173829\" class=\"exercise\">\r\n<div id=\"fs-id1165043173831\" class=\"textbox\">\r\n<p id=\"fs-id1165043173833\"><strong>16.\u00a0<\/strong>[latex]f(x)=2 \\sin (x)+ \\sin (2x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042364505\" class=\"exercise\">\r\n<div id=\"fs-id1165042364507\" class=\"textbox\">\r\n<p id=\"fs-id1165042364510\"><strong>17.\u00a0<\/strong>[latex]f(x)=\\sec^2 (x)+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042364547\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042364547\"]\r\n<p id=\"fs-id1165042364547\">[latex]F(x)=x+ \\tan (x)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043281579\" class=\"exercise\">\r\n<div id=\"fs-id1165043281581\" class=\"textbox\">\r\n<p id=\"fs-id1165043281583\"><strong>18.\u00a0<\/strong>[latex]f(x)= \\sin x \\cos x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043281655\" class=\"exercise\">\r\n<div id=\"fs-id1165043281657\" class=\"textbox\">\r\n<p id=\"fs-id1165043281659\"><strong>19.\u00a0<\/strong>[latex]f(x)= \\sin^2 (x) \\cos (x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043424715\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043424715\"]\r\n<p id=\"fs-id1165043424715\">[latex]F(x)=\\frac{1}{3} \\sin^3 (x)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043424756\" class=\"exercise\">\r\n<div id=\"fs-id1165043424758\" class=\"textbox\">\r\n<p id=\"fs-id1165043424760\"><strong>20.\u00a0<\/strong>[latex]f(x)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042660274\" class=\"exercise\">\r\n<div id=\"fs-id1165042660276\" class=\"textbox\">\r\n<p id=\"fs-id1165042660278\"><strong>21.\u00a0<\/strong>[latex]f(x)=\\frac{1}{2} \\csc^2 (x)+\\frac{1}{x^2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042660326\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042660326\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042660326\"][latex]F(x)=-\\frac{1}{2} \\cot (x)-\\frac{1}{x}+C[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043173691\" class=\"exercise\">\r\n<div id=\"fs-id1165043173693\" class=\"textbox\">\r\n<p id=\"fs-id1165043173696\"><strong>22.\u00a0<\/strong>[latex]f(x)= \\csc x \\cot x+3x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043173769\" class=\"exercise\">\r\n<div id=\"fs-id1165043173771\" class=\"textbox\">\r\n<p id=\"fs-id1165043173773\"><strong>23.\u00a0<\/strong>[latex]f(x)=4 \\csc x \\cot x- \\sec x \\tan x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<div class=\"textbox shaded\">[reveal-answer q=\"93930\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"93930\"][latex]F(x)=\u2212 \\sec x-4 \\csc x+C[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043252042\" class=\"exercise\">\r\n<div id=\"fs-id1165043252044\" class=\"textbox\">\r\n<p id=\"fs-id1165043252046\"><strong>24.\u00a0<\/strong>[latex]f(x)=8 \\sec x( \\sec x-4 \\tan x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042676267\" class=\"exercise\">\r\n<div id=\"fs-id1165042676269\" class=\"textbox\">\r\n<p id=\"fs-id1165042676271\"><strong>25.\u00a0<\/strong>[latex]f(x)=\\frac{1}{2}e^{-4x}+ \\sin x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042676311\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042676311\"]\r\n<p id=\"fs-id1165042676311\">[latex]F(x)=-\\frac{1}{8}e^{-4x}- \\cos x+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042681088\">For the following exercises, evaluate the integral.<\/p>\r\n\r\n<div id=\"fs-id1165042681091\" class=\"exercise\">\r\n<div id=\"fs-id1165042681094\" class=\"textbox\">\r\n<p id=\"fs-id1165042681096\"><strong>26.\u00a0<\/strong>[latex]\\int (-1) dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042681137\" class=\"exercise\">\r\n<div id=\"fs-id1165042681139\" class=\"textbox\">\r\n<p id=\"fs-id1165042681142\"><strong>27.\u00a0<\/strong>[latex]\\int \\sin x dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<div class=\"textbox shaded\">\r\n<div id=\"fs-id1165042681137\" class=\"exercise\">\r\n<div class=\"solution\">\r\n<p id=\"fs-id1165042681164\">[reveal-answer q=\"436260\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"436260\"][latex]\u2212 \\cos x+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042681181\" class=\"exercise\">\r\n<div id=\"fs-id1165042681183\" class=\"textbox\">\r\n<p id=\"fs-id1165042681185\"><strong>28.\u00a0<\/strong>[latex]\\int (4x+\\sqrt{x}) dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042461202\" class=\"exercise\">\r\n<div id=\"fs-id1165042461204\" class=\"textbox\">\r\n<p id=\"fs-id1165042461206\"><strong>29.\u00a0<\/strong>[latex]\\int \\frac{3x^2+2}{x^2} dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042461244\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042461244\"]\r\n<p id=\"fs-id1165042461244\">[latex]3x-\\frac{2}{x}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042374847\" class=\"exercise\">\r\n<div id=\"fs-id1165042374849\" class=\"textbox\">\r\n<p id=\"fs-id1165042374851\"><strong>30.\u00a0<\/strong>[latex]\\int (\\sec x \\tan x+4x) dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042374915\" class=\"exercise\">\r\n<div id=\"fs-id1165042374917\" class=\"textbox\">\r\n<p id=\"fs-id1165042374920\"><strong>31.\u00a0<\/strong>[latex]\\int (4\\sqrt{x}+\\sqrt[4]{x}) dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042374957\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042374957\"]\r\n<p id=\"fs-id1165042374957\">[latex]\\frac{8}{3}x^{3\/2}+\\frac{4}{5}x^{5\/4}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042327522\" class=\"exercise\">\r\n<div id=\"fs-id1165042327524\" class=\"textbox\">\r\n<p id=\"fs-id1165042327526\"><strong>32.\u00a0<\/strong>[latex]\\int (x^{-1\/3}-x^{2\/3}) dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042700339\" class=\"exercise\">\r\n<div id=\"fs-id1165042700341\" class=\"textbox\">\r\n<p id=\"fs-id1165042700343\"><strong>33.\u00a0<\/strong>[latex]\\int \\frac{14x^3+2x+1}{x^3} dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042700388\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042700388\"]\r\n<p id=\"fs-id1165042700388\">[latex]14x-\\frac{2}{x}-\\frac{1}{2x^2}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042700424\" class=\"exercise\">\r\n<div id=\"fs-id1165042700427\" class=\"textbox\">\r\n<p id=\"fs-id1165042700429\"><strong>34.\u00a0<\/strong>[latex]\\int (e^x+e^{\u2212x}) dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042668727\">For the following exercises, solve the initial value problem.<\/p>\r\n\r\n<div id=\"fs-id1165042668731\" class=\"exercise\">\r\n<div id=\"fs-id1165042668733\" class=\"textbox\">\r\n<p id=\"fs-id1165042668735\"><strong>35.\u00a0<\/strong>[latex]f^{\\prime}(x)=x^{-3}, \\, f(1)=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042668780\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042668780\"]\r\n<p id=\"fs-id1165042668780\">[latex]f(x)=-\\frac{1}{2x^2}+\\frac{3}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042668819\" class=\"exercise\">\r\n<div id=\"fs-id1165042668821\" class=\"textbox\">\r\n<p id=\"fs-id1165042668823\"><strong>36.\u00a0<\/strong>[latex]f^{\\prime}(x)=\\sqrt{x}+x^2, \\, f(0)=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042315622\" class=\"exercise\">\r\n<div id=\"fs-id1165042315624\" class=\"textbox\">\r\n<p id=\"fs-id1165042315626\"><strong>37.\u00a0<\/strong>[latex]f^{\\prime}(x)= \\cos x+ \\sec^2 (x), \\, f(\\frac{\\pi}{4})=2+\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042684159\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042684159\"]\r\n<p id=\"fs-id1165042684159\">[latex]f(x)= \\sin x+ \\tan x+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042684192\" class=\"exercise\">\r\n<div id=\"fs-id1165042684194\" class=\"textbox\">\r\n<p id=\"fs-id1165042684196\"><strong>38.\u00a0<\/strong>[latex]f^{\\prime}(x)=x^3-8x^2+16x+1, \\, f(0)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042684316\" class=\"exercise\">\r\n<div id=\"fs-id1165042684318\" class=\"textbox\">\r\n<p id=\"fs-id1165042684320\"><strong>39.\u00a0<\/strong>[latex]f^{\\prime}(x)=\\frac{2}{x^2}-\\frac{x^2}{2}, \\, f(1)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043422382\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043422382\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043422382\"][latex]f(x)=-\\frac{1}{6}x^3-\\frac{2}{x}+\\frac{13}{6}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165043422431\">For the following exercises, find two possible functions [latex]f[\/latex] given the second- or third-order derivatives.<\/p>\r\n\r\n<div id=\"fs-id1165043422439\" class=\"exercise\">\r\n<div id=\"fs-id1165043422441\" class=\"textbox\">\r\n<p id=\"fs-id1165043422443\"><strong>40.\u00a0<\/strong>[latex]f^{\\prime \\prime}(x)=x^2+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042326003\" class=\"exercise\">\r\n<div id=\"fs-id1165042326005\" class=\"textbox\">\r\n<p id=\"fs-id1165042326007\"><strong>41.\u00a0<\/strong>[latex]f^{\\prime \\prime}(x)=e^{\u2212x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042326038\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042326038\"]\r\n<p id=\"fs-id1165042326038\">Answers may vary; one possible answer is [latex]f(x)=e^{\u2212x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042326066\" class=\"exercise\">\r\n<div id=\"fs-id1165042326068\" class=\"textbox\">\r\n<p id=\"fs-id1165042326070\"><strong>42.\u00a0<\/strong>[latex]f^{\\prime \\prime}(x)=1+x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042326139\" class=\"exercise\">\r\n<div id=\"fs-id1165042326141\" class=\"textbox\">\r\n<p id=\"fs-id1165042326144\"><strong>43.\u00a0<\/strong>[latex]f^{\\prime \\prime \\prime}(x)= \\cos x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<div class=\"textbox shaded\">\r\n[reveal-answer q=\"671289\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"671289\"]Answers may vary; one possible answer is [latex]f(x)=\u2212 \\sin x[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042667012\" class=\"exercise\">\r\n<div id=\"fs-id1165042667014\" class=\"textbox\">\r\n<p id=\"fs-id1165042667017\"><strong>44.\u00a0<\/strong>[latex]f^{\\prime \\prime \\prime}(x)=8e^{-2x}- \\sin x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042667092\" class=\"exercise\">\r\n<div id=\"fs-id1165042667094\" class=\"textbox\">\r\n<p id=\"fs-id1165042667097\"><strong>45.\u00a0<\/strong>A car is being driven at a rate of 40 mph when the brakes are applied. The car decelerates at a constant rate of 10 ft\/sec<sup>2<\/sup>. How long before the car stops?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042667117\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042667117\"]\r\n<p id=\"fs-id1165042667117\">5.867 sec<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042667126\" class=\"exercise\">\r\n<div id=\"fs-id1165042667129\" class=\"textbox\">\r\n<p id=\"fs-id1165042667131\"><strong>46.\u00a0<\/strong>In the preceding problem, calculate how far the car travels in the time it takes to stop.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042667147\" class=\"exercise\">\r\n<div id=\"fs-id1165042667149\" class=\"textbox\">\r\n<p id=\"fs-id1165042667151\"><strong>47.\u00a0<\/strong>You are merging onto the freeway, accelerating at a constant rate of 12 ft\/sec<sup>2<\/sup>. How long does it take you to reach merging speed at 60 mph?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042667171\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042667171\"]\r\n<p id=\"fs-id1165042667171\">7.333 sec<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042667181\" class=\"exercise\">\r\n<div id=\"fs-id1165042498516\" class=\"textbox\">\r\n<p id=\"fs-id1165042498518\"><strong>48.\u00a0<\/strong>Based on the previous problem, how far does the car travel to reach merging speed?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042498534\" class=\"exercise\">\r\n<div id=\"fs-id1165042498536\" class=\"textbox\">\r\n<p id=\"fs-id1165042498538\"><strong>49.\u00a0<\/strong>A car company wants to ensure its newest model can stop in 8 sec when traveling at 75 mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042498554\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042498554\"]\r\n<p id=\"fs-id1165042498554\">13.75 ft\/sec<sup>2<\/sup><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042498566\" class=\"exercise\">\r\n<div id=\"fs-id1165042498568\" class=\"textbox\">\r\n<p id=\"fs-id1165042498570\"><strong>50.\u00a0<\/strong>A car company wants to ensure its newest model can stop in less than 450 ft when traveling at 60 mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042498599\">For the following exercises, find the antiderivative of the function, assuming [latex]F(0)=0[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165042498620\" class=\"exercise\">\r\n<div id=\"fs-id1165042498622\" class=\"textbox\">\r\n<p id=\"fs-id1165042498624\"><strong>51. [T]\u00a0<\/strong>[latex]f(x)=x^2+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042498658\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042498658\"]\r\n<p id=\"fs-id1165042498658\">[latex]F(x)=\\frac{1}{3}x^3+2x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042498693\" class=\"exercise\">\r\n<div id=\"fs-id1165042498695\" class=\"textbox\">\r\n<p id=\"fs-id1165042498697\"><strong>52. [T]\u00a0<\/strong>[latex]f(x)=4x-\\sqrt{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042631784\" class=\"exercise\">\r\n<div id=\"fs-id1165042631786\" class=\"textbox\">\r\n<p id=\"fs-id1165042631788\"><strong>53. [T]\u00a0<\/strong>[latex]f(x)= \\sin x+2x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<div class=\"textbox shaded\">\r\n<div id=\"fs-id1165042631784\" class=\"exercise\">\r\n<div class=\"solution\">\r\n<p id=\"fs-id1165042631822\">[reveal-answer q=\"814324\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"814324\"][latex]F(x)=x^2- \\cos x+1[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042631857\" class=\"exercise\">\r\n<div id=\"fs-id1165042631859\" class=\"textbox\">\r\n<p id=\"fs-id1165042631861\"><strong>54. [T]\u00a0<\/strong>[latex]f(x)=e^x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042631917\" class=\"exercise\">\r\n<div id=\"fs-id1165042631920\" class=\"textbox\">\r\n<p id=\"fs-id1165042631922\"><strong>55. [T]\u00a0<\/strong>[latex]f(x)=\\frac{1}{(x+1)^2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042418084\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042418084\"]\r\n<p id=\"fs-id1165042418084\">[latex]F(x)=-\\frac{1}{x+1}+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042418126\" class=\"exercise\">\r\n<div id=\"fs-id1165042418128\" class=\"textbox\">\r\n<p id=\"fs-id1165042418130\"><strong>56. [T]\u00a0<\/strong>[latex]f(x)=e^{-2x}+3x^2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042418221\">For the following exercises, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false.<\/p>\r\n\r\n<div id=\"fs-id1165042418225\" class=\"exercise\">\r\n<div id=\"fs-id1165042418227\" class=\"textbox\">\r\n<p id=\"fs-id1165042418229\"><strong>57.\u00a0<\/strong>If [latex]f(x)[\/latex] is the antiderivative of [latex]v(x)[\/latex], then [latex]2f(x)[\/latex] is the antiderivative of [latex]2v(x)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042518929\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042518929\"]\r\n<p id=\"fs-id1165042518929\">True<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042518934\" class=\"exercise\">\r\n<div id=\"fs-id1165042518937\" class=\"textbox\">\r\n<p id=\"fs-id1165042518939\"><strong>58.\u00a0<\/strong>If [latex]f(x)[\/latex] is the antiderivative of [latex]v(x)[\/latex], then [latex]f(2x)[\/latex] is the antiderivative of [latex]v(2x)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042519011\" class=\"exercise\">\r\n<div id=\"fs-id1165042519013\" class=\"textbox\">\r\n<p id=\"fs-id1165042519016\"><strong>59.\u00a0<\/strong>If [latex]f(x)[\/latex] is the antiderivative of [latex]v(x)[\/latex], then [latex]f(x)+1[\/latex] is the antiderivative of [latex]v(x)+1[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042519085\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042519085\"]\r\n<p id=\"fs-id1165042519085\">False<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042519090\" class=\"exercise\">\r\n<div id=\"fs-id1165042519092\" class=\"textbox\">\r\n<p id=\"fs-id1165042519094\"><strong>60.\u00a0<\/strong>If [latex]f(x)[\/latex] is the antiderivative of [latex]v(x)[\/latex], then [latex](f(x))^2[\/latex] is the antiderivative of [latex](v(x))^2[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165042617549\" class=\"definition\">\r\n \t<dt>antiderivative<\/dt>\r\n \t<dd id=\"fs-id1165042617555\">a function [latex]F[\/latex] such that [latex]F^{\\prime}(x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex] is an antiderivative of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042617603\" class=\"definition\">\r\n \t<dt>indefinite integral<\/dt>\r\n \t<dd id=\"fs-id1165042617608\">the most general antiderivative of [latex]f(x)[\/latex] is the indefinite integral of [latex]f[\/latex]; we use the notation [latex]\\int f(x) dx[\/latex] to denote the indefinite integral of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042617659\" class=\"definition\">\r\n \t<dt>initial value problem<\/dt>\r\n \t<dd id=\"fs-id1165042617665\">a problem that requires finding a function [latex]y[\/latex] that satisfies the differential equation [latex]\\frac{dy}{dx}=f(x)[\/latex] together with the initial condition [latex]y(x_0)=y_0[\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Find the general antiderivative of a given function.<\/li>\n<li>Explain the terms and notation used for an indefinite integral.<\/li>\n<li>State the power rule for integrals.<\/li>\n<li>Use antidifferentiation to solve simple initial-value problems.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165042951286\">At this point, we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: Given a function [latex]f[\/latex], how do we find a function with the derivative [latex]f[\/latex] and why would we be interested in such a function?<\/p>\n<p id=\"fs-id1165043307884\">We answer the first part of this question by defining antiderivatives. The <strong>antiderivative<\/strong> of a function [latex]f[\/latex] is a function with a derivative [latex]f[\/latex]. Why are we interested in antiderivatives? The need for antiderivatives arises in many situations, and we look at various examples throughout the remainder of the text. Here we examine one specific example that involves rectilinear motion. In our examination in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction-4\/\">Derivatives<\/a> of rectilinear motion, we showed that given a position function [latex]s(t)[\/latex] of an object, then its velocity function [latex]v(t)[\/latex] is the derivative of [latex]s(t)[\/latex]\u2014that is, [latex]v(t)=s^{\\prime}(t)[\/latex]. Furthermore, the acceleration [latex]a(t)[\/latex] is the derivative of the velocity [latex]v(t)[\/latex]\u2014that is, [latex]a(t)=v^{\\prime}(t)=s^{\\prime \\prime}(t)[\/latex]. Now suppose we are given an acceleration function [latex]a[\/latex], but not the velocity function [latex]v[\/latex] or the position function [latex]s[\/latex]. Since [latex]a(t)=v^{\\prime}(t)[\/latex], determining the velocity function requires us to find an antiderivative of the acceleration function. Then, since [latex]v(t)=s^{\\prime}(t)[\/latex], determining the position function requires us to find an antiderivative of the velocity function. Rectilinear motion is just one case in which the need for antiderivatives arises. We will see many more examples throughout the remainder of the text. For now, let\u2019s look at the terminology and notation for antiderivatives, and determine the antiderivatives for several types of functions. We examine various techniques for finding antiderivatives of more complicated functions in the second volume of this text (<a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/HTmjSAcf@2.46:Z4WWhBaa@3\/Introduction\">Introduction to Techniques of Integration<\/a>).<\/p>\n<div id=\"fs-id1165043404679\" class=\"bc-section section\">\n<h1>The Reverse of Differentiation<\/h1>\n<p id=\"fs-id1165043323795\">At this point, we know how to find derivatives of various functions. We now ask the opposite question. Given a function [latex]f[\/latex], how can we find a function with derivative [latex]f[\/latex]? If we can find a function [latex]F[\/latex] with derivative [latex]f[\/latex], we call [latex]F[\/latex] an antiderivative of [latex]f[\/latex].<\/p>\n<div id=\"fs-id1165042478105\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1165042887564\">A function [latex]F[\/latex] is an antiderivative of the function [latex]f[\/latex] if<\/p>\n<div id=\"fs-id1165042945928\" class=\"equation unnumbered\">[latex]F^{\\prime}(x)=f(x)[\/latex]<\/div>\n<p id=\"fs-id1165042964886\">for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\n<\/div>\n<p>Consider the function [latex]f(x)=2x[\/latex]. Knowing the power rule of differentiation, we conclude that [latex]F(x)=x^2[\/latex] is an antiderivative of [latex]f[\/latex] since [latex]F^{\\prime}(x)=2x[\/latex]. Are there any other antiderivatives of [latex]f[\/latex]? Yes; since the derivative of any constant [latex]C[\/latex] is zero, [latex]x^2+C[\/latex] is also an antiderivative of [latex]2x[\/latex]. Therefore, [latex]x^2+5[\/latex] and [latex]x^{2}-\\sqrt{2}[\/latex] are also antiderivatives. Are there any others that are not of the form [latex]x^2+C[\/latex] for some constant [latex]C[\/latex]? The answer is no. From Corollary 2 of the Mean Value Theorem, we know that if [latex]F[\/latex] and [latex]G[\/latex] are differentiable functions such that [latex]F^{\\prime}(x)=G^{\\prime}(x)[\/latex], then [latex]F(x)-G(x)=C[\/latex] for some constant [latex]C[\/latex]. This fact leads to the following important theorem.<\/p>\n<div class=\"textbox key-takeaways theorem\">\n<h3>General Form of an Antiderivative<\/h3>\n<p id=\"fs-id1165042884523\">Let [latex]F[\/latex] be an antiderivative of [latex]f[\/latex] over an interval [latex]I[\/latex]. Then,<\/p>\n<ol id=\"fs-id1165043009392\">\n<li>for each constant [latex]C[\/latex], the function [latex]F(x)+C[\/latex] is also an antiderivative of [latex]f[\/latex] over [latex]I[\/latex];<\/li>\n<li>if [latex]G[\/latex] is an antiderivative of [latex]f[\/latex] over [latex]I[\/latex], there is a constant [latex]C[\/latex] for which [latex]G(x)=F(x)+C[\/latex] over [latex]I[\/latex].<\/li>\n<\/ol>\n<p id=\"fs-id1165042987370\">In other words, the most general form of the antiderivative of [latex]f[\/latex] over [latex]I[\/latex] is [latex]F(x)+C[\/latex].<\/p>\n<\/div>\n<p>We use this fact and our knowledge of derivatives to find all the antiderivatives for several functions.<\/p>\n<div id=\"fs-id1165042616955\" class=\"textbox examples\">\n<h3>Finding Antiderivatives<\/h3>\n<div id=\"fs-id1165042984701\" class=\"exercise\">\n<div id=\"fs-id1165043431382\" class=\"textbox\">\n<p>For each of the following functions, find all antiderivatives.<\/p>\n<ol id=\"fs-id1165043115403\" style=\"list-style-type: lower-alpha\">\n<li>[latex]f(x)=3x^2[\/latex]<\/li>\n<li>[latex]f(x)=\\frac{1}{x}[\/latex]<\/li>\n<li>[latex]f(x)= \\cos x[\/latex]<\/li>\n<li>[latex]f(x)=e^x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"solution\">\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q46129\">Show Answer<\/span><\/p>\n<div id=\"q46129\" class=\"hidden-answer\" style=\"display: none\">a. Because<\/p>\n<p>[latex]\\frac{d}{dx}(x^3)=3x^2[\/latex]<\/p>\n<p>then [latex]F(x)=x^3[\/latex] is an antiderivative of [latex]3x^2[\/latex]. Therefore, every antiderivative of [latex]3x^2[\/latex] is of the form [latex]x^3+C[\/latex] for some constant [latex]C[\/latex], and every function of the form [latex]x^3+C[\/latex] is an antiderivative of [latex]3x^2[\/latex].<\/p>\n<p>b. Let [latex]f(x)=\\ln |x|[\/latex]. For [latex]x>0, \\, f(x)=\\ln (x)[\/latex] and<\/p>\n<p>[latex]\\frac{d}{dx}(\\ln x)=\\frac{1}{x}[\/latex].<\/p>\n<p>For [latex]x<0, \\, f(x)=\\ln (\u2212x)[\/latex] and\n\n[latex]\\frac{d}{dx}(\\ln (\u2212x))=-\\frac{1}{\u2212x}=\\frac{1}{x}[\/latex].\n\nTherefore,\n\n[latex]\\frac{d}{dx}(\\ln |x|)=\\frac{1}{x}[\/latex].\n\nThus, [latex]F(x)=\\ln |x|[\/latex] is an antiderivative of [latex]\\frac{1}{x}[\/latex]. Therefore, every antiderivative of [latex]\\frac{1}{x}[\/latex] is of the form [latex]\\ln |x|+C[\/latex] for some constant [latex]C[\/latex] and every function of the form [latex]\\ln |x|+C[\/latex] is an antiderivative of [latex]\\frac{1}{x}[\/latex].\n\nc. We have\n\n[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex],\n\nso [latex]F(x)= \\sin x[\/latex] is an antiderivative of [latex]\\cos x[\/latex]. Therefore, every antiderivative of [latex]\\cos x[\/latex] is of the form [latex]\\sin x+C[\/latex] for some constant [latex]C[\/latex] and every function of the form [latex]\\sin x+C[\/latex] is an antiderivative of [latex]\\cos x[\/latex].\n\nd. Since\n\n[latex]\\frac{d}{dx}(e^x)=e^x[\/latex],\n\nthen [latex]F(x)=e^x[\/latex] is an antiderivative of [latex]e^x[\/latex]. Therefore, every antiderivative of [latex]e^x[\/latex] is of the form [latex]e^x+C[\/latex] for some constant [latex]C[\/latex] and every function of the form [latex]e^x+C[\/latex] is an antiderivative of [latex]e^x[\/latex].<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043353933\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042966615\" class=\"exercise\">\n<div id=\"fs-id1165043194525\" class=\"textbox\">\n<p id=\"fs-id1165043353234\">Find all antiderivatives of [latex]f(x)= \\sin x[\/latex].<\/p>\n<\/div>\n<div class=\"solution\">\n<div class=\"textbox shaded\">\n<div class=\"solution\">\n<p id=\"fs-id1165042373760\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q314667\">Show Answer<\/span><\/p>\n<div id=\"q314667\" class=\"hidden-answer\" style=\"display: none\">[latex]\u2212\\cos x+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1165043379984\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p>What function has a derivative of [latex]\\sin x[\/latex]?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042332032\" class=\"bc-section section\">\n<h1>Indefinite Integrals<\/h1>\n<p id=\"fs-id1165043344704\">We now look at the formal notation used to represent antiderivatives and examine some of their properties. These properties allow us to find antiderivatives of more complicated functions. Given a function [latex]f[\/latex], we use the notation [latex]f^{\\prime}(x)[\/latex] or [latex]\\frac{df}{dx}[\/latex] to denote the derivative of [latex]f[\/latex]. Here we introduce notation for antiderivatives. If [latex]F[\/latex] is an antiderivative of [latex]f[\/latex], we say that [latex]F(x)+C[\/latex] is the most general antiderivative of [latex]f[\/latex] and write<\/p>\n<div class=\"equation unnumbered\">[latex]\\int f(x) dx=F(x)+C[\/latex].<\/div>\n<p id=\"fs-id1165042959838\">The symbol [latex]\\int[\/latex] is called an <em>integral sign<\/em>, and [latex]\\int f(x) dx[\/latex] is called the<strong> indefinite integral<\/strong> of [latex]f[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1165043393369\">Given a function [latex]f[\/latex], the indefinite integral of [latex]f[\/latex], denoted<\/p>\n<div class=\"equation unnumbered\">[latex]\\int f(x) dx[\/latex],<\/div>\n<p>is the most general antiderivative of [latex]f[\/latex]. If [latex]F[\/latex] is an antiderivative of [latex]f[\/latex], then<\/p>\n<div id=\"fs-id1165043119692\" class=\"equation unnumbered\">[latex]\\int f(x) dx=F(x)+C[\/latex].<\/div>\n<p id=\"fs-id1165043096049\">The expression [latex]f(x)[\/latex] is called the <em>integrand<\/em> and the variable [latex]x[\/latex] is the <em>variable of integration<\/em>.<\/p>\n<\/div>\n<p id=\"fs-id1165042558766\">Given the terminology introduced in this definition, the act of finding the antiderivatives of a function [latex]f[\/latex] is usually referred to as <em>integrating<\/em> [latex]f[\/latex].<\/p>\n<p id=\"fs-id1165042936505\">For a function [latex]f[\/latex] and an antiderivative [latex]F[\/latex], the functions [latex]F(x)+C[\/latex], where [latex]C[\/latex] is any real number, is often referred to as <em>the family of antiderivatives of<\/em> [latex]f[\/latex]. For example, since [latex]x^2[\/latex] is an antiderivative of [latex]2x[\/latex] and any antiderivative of [latex]2x[\/latex] is of the form [latex]x^2+C[\/latex], we write<\/p>\n<div id=\"fs-id1165043327744\" class=\"equation unnumbered\">[latex]\\int 2x dx=x^2+C[\/latex].<\/div>\n<p id=\"fs-id1165043097500\">The collection of all functions of the form [latex]x^2+C[\/latex], where [latex]C[\/latex] is any real number, is known as the <em>family of antiderivatives of<\/em> [latex]2x[\/latex].\u00a0<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_10_001\">(Figure)<\/a> shows a graph of this family of antiderivatives.<\/p>\n<div id=\"CNX_Calc_Figure_04_10_001\" class=\"wp-caption aligncenter\">\n<div style=\"width: 656px\" 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\/11211358\/CNX_Calc_Figure_04_10_001.jpg\" alt=\"The graphs for y = x2 + 2, y = x2 + 1, y = x2, y = x2 \u2212 1, and y = x2 \u2212 2 are shown.\" width=\"646\" height=\"575\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1.<\/strong> The family of antiderivatives of [latex]2x[\/latex] consists of all functions of the form [latex]x^2+C[\/latex], where [latex]C[\/latex] is any real number.<\/p>\n<\/div>\n<\/div>\n<p>For some functions, evaluating indefinite integrals follows directly from properties of derivatives. For example, for [latex]n \\ne \u22121[\/latex],<\/p>\n<div id=\"fs-id1165043350427\" class=\"equation unnumbered\">[latex]\\int x^n dx=\\frac{x^{n+1}}{n+1}+C[\/latex],<\/div>\n<p id=\"fs-id1165043015098\">which comes directly from<\/p>\n<div id=\"fs-id1165043036022\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}(\\frac{x^{n+1}}{n+1})=(n+1)\\frac{x^n}{n+1}=x^n[\/latex].<\/div>\n<p id=\"fs-id1165042343281\">This fact is known as <em>the power rule for integrals<\/em>.<\/p>\n<div id=\"fs-id1165043041347\" class=\"textbox key-takeaways theorem\">\n<h3>Power Rule for Integrals<\/h3>\n<p id=\"fs-id1165042514785\">For [latex]n \\ne \u22121[\/latex],<\/p>\n<div id=\"fs-id1165043250161\" class=\"equation unnumbered\">[latex]\\int x^n dx=\\frac{x^{n+1}}{n+1}+C[\/latex].<\/div>\n<\/div>\n<p id=\"fs-id1165043385541\">Evaluating indefinite integrals for some other functions is also a straightforward calculation. The following table lists the indefinite integrals for several common functions. A more complete list appears in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/back-matter\/table-of-derivatives\/\">Appendix B<\/a>.<\/p>\n<table summary=\"This is a table with two columns and fourteen rows, titled \u201cIntegration Formulas.\u201d The first row is a header row, and labels column one \u201cDifferentiation Formula\u201d and column two \u201cIndefinite Integral.\u201d The second row reads d\/dx (k) = 0, the integral of kdx = the integral of kx^0dx = kx + C. The third row reads d\/dx(x^n) = nx^(x-1), the integral of x^ndn = (x^n+1)\/(n+1) + C for n is not equal to negative 1. The fourth row reads d\/dx(ln(the absolute value of x))=1\/x, the integral of (1\/x)dx = ln(the absolute value of x) + C. The fifth row reads d\/dx(e^x) = e^x, the integral of e^xdx = e^x + C. The sixth row reads d\/dx(sinx) = cosx, the integral of cosxdx = sinx + C. The seventh row reads d\/dx(cosx) = negative sinx, the integral of sinxdx = negative cosx + C. The eighth row reads d\/dx(tanx) = sec squared x, the integral of sec squared xdx = tanx + C. The ninth row reads d\/dx(cscx) = negative cscxcotx, the integral of cscxcotxdx = negative cscx + C. The tenth row reads d\/dx(secx) = secxtanx, the integral of secxtanxdx = secx + C. The eleventh row reads d\/dx(cotx) = negative csc squared x, the integral of csc squared xdx = negative cot x + C. The twelfth row reads d\/dx(sin^-1(x)) = 1\/the square root of (1 \u2013 x^2), the integral of 1\/(the square root of (x^2 \u2013 1) = sin^-1(x) + C. The thirteenth row reads d\/dx (tan^-1(x)) = 1\/(1 + x^2), the integral of 1\/(1 + x^2)dx = tan^-1(x) + C. The fourteenth row reads d\/dx(sec^-1(the absolute value of x)) = 1\/x(the square root of x^2 \u2013 1), the integral of 1\/x(the square root of x^2 \u2013 1)dx = sec^-1(the absolute value of x) + C.\">\n<caption>Integration Formulas<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>Differentiation Formula<\/th>\n<th>Indefinite Integral<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(k)=0[\/latex]<\/td>\n<td>[latex]\\int kdx=\\int kx^0 dx=kx+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(x^n)=nx^{n-1}[\/latex]<\/td>\n<td>[latex]\\int x^n dx=\\frac{x^{n+1}}{n+1}+C[\/latex] for [latex]n\\ne \u22121[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\ln |x|)=\\frac{1}{x}[\/latex]<\/td>\n<td>[latex]\\int \\frac{1}{x}dx=\\ln |x|+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(e^x)=e^x[\/latex]<\/td>\n<td>[latex]\\int e^x dx=e^x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/td>\n<td>[latex]\\int \\cos x dx= \\sin x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\cos x)=\u2212 \\sin x[\/latex]<\/td>\n<td>[latex]\\int \\sin x dx=\u2212 \\cos x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\tan x)= \\sec^2 x[\/latex]<\/td>\n<td>[latex]\\int \\sec^2 x dx= \\tan x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/td>\n<td>[latex]\\int \\csc x \\cot x dx=\u2212\\csc x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/td>\n<td>[latex]\\int \\sec x \\tan x dx= \\sec x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/td>\n<td>[latex]\\int \\csc^2 x dx=\u2212\\cot x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}( \\sin^{-1} x)=\\frac{1}{\\sqrt{1-x^2}}[\/latex]<\/td>\n<td>[latex]\\int \\frac{1}{\\sqrt{1-x^2}} dx= \\sin^{-1} x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\tan^{-1} x)=\\frac{1}{1+x^2}[\/latex]<\/td>\n<td>[latex]\\int \\frac{1}{1+x^2} dx= \\tan^{-1} x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\sec^{-1} |x|)=\\frac{1}{x\\sqrt{x^2-1}}[\/latex]<\/td>\n<td>[latex]\\int \\frac{1}{x\\sqrt{x^2-1}} dx= \\sec^{-1} |x|+C[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042328678\">From the definition of indefinite integral of [latex]f[\/latex], we know<\/p>\n<div id=\"fs-id1165042373302\" class=\"equation unnumbered\">[latex]\\int f(x) dx=F(x)+C[\/latex]<\/div>\n<p id=\"fs-id1165042322393\">if and only if [latex]F[\/latex] is an antiderivative of [latex]f[\/latex]. Therefore, when claiming that<\/p>\n<div id=\"fs-id1165042472053\" class=\"equation unnumbered\">[latex]\\int f(x) dx=F(x)+C[\/latex]<\/div>\n<p id=\"fs-id1165042375807\">it is important to check whether this statement is correct by verifying that [latex]F^{\\prime}(x)=f(x)[\/latex].<\/p>\n<div id=\"fs-id1165043393824\" class=\"textbox examples\">\n<h3>Verifying an Indefinite Integral<\/h3>\n<div id=\"fs-id1165043393826\" class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1165043428255\">Each of the following statements is of the form [latex]\\int f(x) dx=F(x)+C[\/latex]. Verify that each statement is correct by showing that [latex]F^{\\prime}(x)=f(x)[\/latex].<\/p>\n<ol id=\"fs-id1165043281729\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\int (x+e^x) dx=\\frac{x^2}{2}+e^x+C[\/latex]<\/li>\n<li>[latex]\\int xe^xdx=xe^x-e^x+C[\/latex]<\/li>\n<\/ol>\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-id1165042710847\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042710847\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165042710847\" style=\"list-style-type: lower-alpha\">\n<li>Since\n<div id=\"fs-id1165042710856\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}(\\frac{x^2}{2}+e^x+C)=x+e^x[\/latex],<\/div>\n<p>the statement<\/p>\n<div id=\"fs-id1165042319135\" class=\"equation unnumbered\">[latex]\\int (x+e^x)dx=\\frac{x^2}{2}+e^x+C[\/latex]<\/div>\n<p>is correct.<br \/>\nNote that we are verifying an indefinite integral for a sum. Furthermore, [latex]\\frac{x^2}{2}[\/latex] and [latex]e^x[\/latex] are antiderivatives of [latex]x[\/latex] and [latex]e^x[\/latex], respectively, and the sum of the antiderivatives is an antiderivative of the sum. We discuss this fact again later in this section.<\/li>\n<li>Using the product rule, we see that\n<div class=\"equation unnumbered\">[latex]\\frac{d}{dx}(xe^x-e^x+C)=e^x+xe^x-e^x=xe^x[\/latex].<\/div>\n<p>Therefore, the statement<\/p>\n<div id=\"fs-id1165043257181\" class=\"equation unnumbered\">[latex]\\int xe^x dx=xe^x-e^x+C[\/latex]<\/div>\n<p>is correct.<br \/>\nNote that we are verifying an indefinite integral for a product. The antiderivative [latex]xe^x-e^x[\/latex] is not a product of the antiderivatives. Furthermore, the product of antiderivatives, [latex]x^2 e^x\/2[\/latex] is not an antiderivative of [latex]xe^x[\/latex] since<\/p>\n<div id=\"fs-id1165042320874\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}(\\frac{x^2e^x}{2})=xe^x+\\frac{x^2e^x}{2} \\ne xe^x[\/latex].<\/div>\n<p>In general, the product of antiderivatives is not an antiderivative of a product.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043078178\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165043078181\" class=\"exercise\">\n<div id=\"fs-id1165042364598\" class=\"textbox\">\n<p id=\"fs-id1165042364600\">Verify that [latex]\\int x \\cos x dx=x \\sin x+ \\cos x+C[\/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-id1165043257533\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043257533\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043257533\">[latex]\\frac{d}{dx}(x \\sin x+ \\cos x+C)= \\sin x+x \\cos x- \\sin x=x \\cos x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165043219874\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165043219881\">Calculate [latex]\\frac{d}{dx}(x \\sin x+ \\cos x+C)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042318564\">In <a class=\"autogenerated-content\" href=\"#fs-id1165043092431\">(Figure)<\/a>, we listed the indefinite integrals for many elementary functions. Let\u2019s now turn our attention to evaluating indefinite integrals for more complicated functions. For example, consider finding an antiderivative of a sum [latex]f+g[\/latex]. In <a class=\"autogenerated-content\" href=\"#fs-id1165043393824\">(Figure)<\/a>a. we showed that an antiderivative of the sum [latex]x+e^x[\/latex] is given by the sum [latex](\\frac{x^2}{2})+e^x[\/latex]\u2014that is, an antiderivative of a sum is given by a sum of antiderivatives. This result was not specific to this example. In general, if [latex]F[\/latex] and [latex]G[\/latex] are antiderivatives of any functions [latex]f[\/latex] and [latex]g[\/latex], respectively, then<\/p>\n<div class=\"equation unnumbered\">[latex]\\frac{d}{dx}(F(x)+G(x))=F^{\\prime}(x)+G^{\\prime}(x)=f(x)+g(x)[\/latex].<\/div>\n<p id=\"fs-id1165043372671\">Therefore, [latex]F(x)+G(x)[\/latex] is an antiderivative of [latex]f(x)+g(x)[\/latex] and we have<\/p>\n<div class=\"equation unnumbered\">[latex]\\int (f(x)+g(x)) dx=F(x)+G(x)+C[\/latex].<\/div>\n<p id=\"fs-id1165043174082\">Similarly,<\/p>\n<div id=\"fs-id1165043174085\" class=\"equation unnumbered\">[latex]\\int (f(x)-g(x)) dx=F(x)-G(x)+C[\/latex].<\/div>\n<p id=\"fs-id1165042383134\">In addition, consider the task of finding an antiderivative of [latex]kf(x)[\/latex], where [latex]k[\/latex] is any real number. Since<\/p>\n<div class=\"equation unnumbered\">[latex]\\frac{d}{dx}(kf(x))=k\\frac{d}{dx}F(x)=kF^{\\prime}(x)[\/latex]<\/div>\n<p id=\"fs-id1165043222034\">for any real number [latex]k[\/latex], we conclude that<\/p>\n<div id=\"fs-id1165042383898\" class=\"equation unnumbered\">[latex]\\int kf(x) dx=kF(x)+C[\/latex].<\/div>\n<p id=\"fs-id1165043425482\">These properties are summarized next.<\/p>\n<div id=\"fs-id1165043425485\" class=\"textbox key-takeaways theorem\">\n<h3>Properties of Indefinite Integrals<\/h3>\n<p id=\"fs-id1165043395041\">Let [latex]F[\/latex] and [latex]G[\/latex] be antiderivatives of [latex]f[\/latex] and [latex]g[\/latex], respectively, and let [latex]k[\/latex] be any real number.<\/p>\n<p id=\"fs-id1165043393659\">Sums and Differences<\/p>\n<div class=\"equation unnumbered\">[latex]\\int (f(x) \\pm g(x)) dx=F(x) \\pm G(x)+C[\/latex]<\/div>\n<p id=\"fs-id1165042328714\">Constant Multiples<\/p>\n<div id=\"fs-id1165042328717\" class=\"equation unnumbered\">[latex]\\int kf(x) dx=kF(x)+C[\/latex]<\/div>\n<\/div>\n<p>From this theorem, we can evaluate any integral involving a sum, difference, or constant multiple of functions with antiderivatives that are known. Evaluating integrals involving products, quotients, or compositions is more complicated (see <a class=\"autogenerated-content\" href=\"#fs-id1165043393824\">(Figure)<\/a>b. for an example involving an antiderivative of a product.) We look at and address integrals involving these more complicated functions in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction-3\/\">Introduction to Integration<\/a>. In the next example, we examine how to use this theorem to calculate the indefinite integrals of several functions.<\/p>\n<div id=\"fs-id1165043248811\" class=\"textbox examples\">\n<h3>Evaluating Indefinite Integrals<\/h3>\n<div id=\"fs-id1165042705917\" class=\"exercise\">\n<div id=\"fs-id1165042705919\" class=\"textbox\">\n<p>Evaluate each of the following indefinite integrals:<\/p>\n<ol style=\"list-style-type: lower-alpha\">\n<li>[latex]\\int (5x^3-7x^2+3x+4) dx[\/latex]<\/li>\n<li>[latex]\\int \\frac{x^2+4\\sqrt[3]{x}}{x} dx[\/latex]<\/li>\n<li>[latex]\\int \\frac{4}{1+x^2} dx[\/latex]<\/li>\n<li>[latex]\\int \\tan x \\cos x dx[\/latex]<\/li>\n<\/ol>\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-id1165042552215\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042552215\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165042552215\" style=\"list-style-type: lower-alpha\">\n<li>Using <a class=\"autogenerated-content\" href=\"#fs-id1165043425485\">(Figure)<\/a>, we can integrate each of the four terms in the integrand separately. We obtain\n<div id=\"fs-id1165042552227\" class=\"equation unnumbered\">[latex]\\int (5x^3-7x^2+3x+4) dx=\\int 5x^3 dx-\\int 7x^2 dx+\\int 3x dx+\\int 4 dx[\/latex].<\/div>\n<p>From the second part of <a class=\"autogenerated-content\" href=\"#fs-id1165043425485\">(Figure)<\/a>, each coefficient can be written in front of the integral sign, which gives<\/p>\n<div id=\"fs-id1165043312575\" class=\"equation unnumbered\">[latex]\\int 5x^3 dx-\\int 7x^2 dx+\\int 3x dx+\\int 4 dx=5\\int x^3 dx-7\\int x^2 dx+3\\int x dx+4\\int 1 dx[\/latex].<\/div>\n<p>Using the power rule for integrals, we conclude that<\/p>\n<div id=\"fs-id1165042407363\" class=\"equation unnumbered\">[latex]\\int (5x^3-7x^2+3x+4) dx=\\frac{5}{4}x^4-\\frac{7}{3}x^3+\\frac{3}{2}x^2+4x+C[\/latex].<\/div>\n<\/li>\n<li>Rewrite the integrand as\n<div id=\"fs-id1165042371846\" class=\"equation unnumbered\">[latex]\\frac{x^2+4\\sqrt[3]{x}}{x}=\\frac{x^2}{x}+\\frac{4\\sqrt[3]{x}}{x}[\/latex].<\/div>\n<p>Then, to evaluate the integral, integrate each of these terms separately. Using the power rule, we have<\/p>\n<div id=\"fs-id1165043427498\" class=\"equation unnumbered\">[latex]\\begin{array}{ll} \\int (x+\\frac{4}{x^{2\/3}}) dx & =\\int x dx+4\\int x^{-2\/3} dx \\\\ & =\\frac{1}{2}x^2+4\\frac{1}{(\\frac{-2}{3})+1}x^{(-2\/3)+1}+C \\\\ & =\\frac{1}{2}x^2+12x^{1\/3}+C \\end{array}[\/latex]<\/div>\n<\/li>\n<li>Using <a class=\"autogenerated-content\" href=\"#fs-id1165043425485\">(Figure)<\/a>, write the integral as\n<div id=\"fs-id1165043348665\" class=\"equation unnumbered\">[latex]4\\int \\frac{1}{1+x^2} dx[\/latex].<\/div>\n<p>Then, use the fact that [latex]\\tan^{-1} (x)[\/latex] is an antiderivative of [latex]\\frac{1}{1+x^2}[\/latex] to conclude that<\/p>\n<div id=\"fs-id1165042374764\" class=\"equation unnumbered\">[latex]\\int \\frac{4}{1+x^2} dx=4 \\tan^{-1} (x)+C[\/latex].<\/div>\n<\/li>\n<li>Rewrite the integrand as\n<div class=\"equation unnumbered\">[latex]\\tan x \\cos x=\\frac{ \\sin x}{ \\cos x} \\cos x= \\sin x[\/latex].<\/div>\n<p>Therefore,<\/p>\n<div id=\"fs-id1165043317182\" class=\"equation unnumbered\">[latex]\\int \\tan x \\cos x dx=\\int \\sin x dx=\u2212 \\cos x+C[\/latex].<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165043426269\" class=\"exercise\">\n<div id=\"fs-id1165043426271\" class=\"textbox\">\n<p>Evaluate [latex]\\int (4x^3-5x^2+x-7) 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-id1165043259694\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043259694\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043259694\">[latex]x^4-\\frac{5}{3}x^3+\\frac{1}{2}x^2-7x+C[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042468208\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042323621\">Integrate each term in the integrand separately, making use of the power rule.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042323629\" class=\"bc-section section\">\n<h1>Initial-Value Problems<\/h1>\n<p id=\"fs-id1165042323634\">We look at techniques for integrating a large variety of functions involving products, quotients, and compositions later in the text. Here we turn to one common use for antiderivatives that arises often in many applications: solving differential equations.<\/p>\n<p id=\"fs-id1165042323639\">A <em>differential equation<\/em> is an equation that relates an unknown function and one or more of its derivatives. The equation<\/p>\n<div id=\"fs-id1165042349924\" class=\"equation\">[latex]\\frac{dy}{dx}=f(x)[\/latex]<\/div>\n<p id=\"fs-id1165042349952\">is a simple example of a differential equation. Solving this equation means finding a function [latex]y[\/latex] with a derivative [latex]f[\/latex]. Therefore, the solutions of <a class=\"autogenerated-content\" href=\"#fs-id1165042349924\">(Figure)<\/a> are the antiderivatives of [latex]f[\/latex]. If [latex]F[\/latex] is one antiderivative of [latex]f[\/latex], every function of the form [latex]y=F(x)+C[\/latex] is a solution of that differential equation. For example, the solutions of<\/p>\n<div id=\"fs-id1165042323537\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=6x^2[\/latex]<\/div>\n<p>are given by<\/p>\n<div id=\"fs-id1165043431000\" class=\"equation unnumbered\">[latex]y=\\int 6x^2 dx=2x^3+C[\/latex].<\/div>\n<p id=\"fs-id1165042407328\">Sometimes we are interested in determining whether a particular solution curve passes through a certain point [latex](x_0,y_0)[\/latex]\u2014that is, [latex]y(x_0)=y_0[\/latex]. The problem of finding a function [latex]y[\/latex] that satisfies a differential equation<\/p>\n<div id=\"fs-id1165042375659\" class=\"equation\">[latex]\\frac{dy}{dx}=f(x)[\/latex]<\/div>\n<p id=\"fs-id1165042368468\">with the additional condition<\/p>\n<div id=\"fs-id1165042368471\" class=\"equation\">[latex]y(x_0)=y_0[\/latex]<\/div>\n<p>is an example of an <strong>initial-value problem<\/strong>. The condition [latex]y(x_0)=y_0[\/latex] is known as an <em>initial condition<\/em>. For example, looking for a function [latex]y[\/latex] that satisfies the differential equation<\/p>\n<div id=\"fs-id1165042545822\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=6x^2[\/latex]<\/div>\n<p id=\"fs-id1165043393696\">and the initial condition<\/p>\n<div id=\"fs-id1165043393699\" class=\"equation unnumbered\">[latex]y(1)=5[\/latex]<\/div>\n<p id=\"fs-id1165043393719\">is an example of an initial-value problem. Since the solutions of the differential equation are [latex]y=2x^3+C[\/latex], to find a function [latex]y[\/latex] that also satisfies the initial condition, we need to find [latex]C[\/latex] such that [latex]y(1)=2(1)^3+C=5[\/latex]. From this equation, we see that [latex]C=3[\/latex], and we conclude that [latex]y=2x^3+3[\/latex] is the solution of this initial-value problem as shown in the following graph.<\/p>\n<div id=\"CNX_Calc_Figure_04_10_002\" class=\"wp-caption aligncenter\">\n<div style=\"width: 651px\" 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\/11211402\/CNX_Calc_Figure_04_10_002.jpg\" alt=\"The graphs for y = 2x3 + 6, y = 2x3 + 3, y = 2x3, and y = 2x3 \u2212 3 are shown.\" width=\"641\" height=\"497\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2.<\/strong> Some of the solution curves of the differential equation [latex]\\frac{dy}{dx}=6x^2[\/latex] are displayed. The function [latex]y=2x^3+3[\/latex] satisfies the differential equation and the initial condition [latex]y(1)=5[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043327648\" class=\"textbox examples\">\n<h3>Solving an Initial-Value Problem<\/h3>\n<div id=\"fs-id1165043327650\" class=\"exercise\">\n<div id=\"fs-id1165043327652\" class=\"textbox\">\n<p id=\"fs-id1165043327658\">Solve the initial-value problem<\/p>\n<div id=\"fs-id1165043327661\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}= \\sin x, \\, y(0)=5[\/latex].<\/div>\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-id1165043430929\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043430929\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043430929\">First we need to solve the differential equation. If [latex]\\frac{dy}{dx}= \\sin x,[\/latex] then<\/p>\n<div id=\"fs-id1165043286683\" class=\"equation unnumbered\">[latex]y=\\int \\sin (x) dx=\u2212 \\cos x+C[\/latex].<\/div>\n<p id=\"fs-id1165042318817\">Next we need to look for a solution [latex]y[\/latex] that satisfies the initial condition. The initial condition [latex]y(0)=5[\/latex] means we need a constant [latex]C[\/latex] such that [latex]\u2212 \\cos x+C=5[\/latex]. Therefore,<\/p>\n<div id=\"fs-id1165043424804\" class=\"equation unnumbered\">[latex]C=5+ \\cos (0)=6[\/latex].<\/div>\n<p id=\"fs-id1165043424835\">The solution of the initial-value problem is [latex]y=\u2212 \\cos x+6[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042327334\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042327337\" class=\"exercise\">\n<div id=\"fs-id1165042327339\" class=\"textbox\">\n<p id=\"fs-id1165042327341\">Solve the initial value problem [latex]\\frac{dy}{dx}=3x^{-2}, \\, y(1)=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-id1165043327482\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043327482\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043327482\">[latex]y=-\\frac{3}{x}+5[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042640833\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042640839\">Find all antiderivatives of [latex]f(x)=3x^{-2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165043174043\">Initial-value problems arise in many applications. Next we consider a problem in which a driver applies the brakes in a car. We are interested in how long it takes for the car to stop. Recall that the velocity function [latex]v(t)[\/latex] is the derivative of a position function [latex]s(t)[\/latex], and the acceleration [latex]a(t)[\/latex] is the derivative of the velocity function. In earlier examples in the text, we could calculate the velocity from the position and then compute the acceleration from the velocity. In the next example we work the other way around. Given an acceleration function, we calculate the velocity function. We then use the velocity function to determine the position function.<\/p>\n<div id=\"fs-id1165042640762\" class=\"textbox examples\">\n<h3>Decelerating Car<\/h3>\n<div id=\"fs-id1165042640764\" class=\"exercise\">\n<div id=\"fs-id1165042640767\" class=\"textbox\">\n<p id=\"fs-id1165042640772\">A car is traveling at the rate of 88 ft\/sec (60 mph) when the brakes are applied. The car begins decelerating at a constant rate of 15 ft\/sec<sup>2<\/sup>.<\/p>\n<ol id=\"fs-id1165043430821\" style=\"list-style-type: lower-alpha\">\n<li>How many seconds elapse before the car stops?<\/li>\n<li>How far does the car travel during that time?<\/li>\n<\/ol>\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-id1165043430836\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043430836\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165043430836\" style=\"list-style-type: lower-alpha\">\n<li>First we introduce variables for this problem. Let [latex]t[\/latex] be the time (in seconds) after the brakes are first applied. Let [latex]a(t)[\/latex] be the acceleration of the car (in feet per seconds squared) at time [latex]t[\/latex]. Let [latex]v(t)[\/latex] be the velocity of the car (in feet per second) at time [latex]t[\/latex]. Let [latex]s(t)[\/latex] be the car\u2019s position (in feet) beyond the point where the brakes are applied at time [latex]t[\/latex].<br \/>\nThe car is traveling at a rate of 88 ft\/sec. Therefore, the initial velocity is [latex]v(0)=88[\/latex] ft\/sec. Since the car is decelerating, the acceleration is<\/p>\n<div id=\"fs-id1165043323894\" class=\"equation unnumbered\">[latex]a(t)=-15[\/latex] ft\/sec<sup>2<\/sup><\/div>\n<p>The acceleration is the derivative of the velocity,<\/p>\n<div id=\"fs-id1165043219161\" class=\"equation unnumbered\">[latex]v^{\\prime}(t)=-15[\/latex].<\/div>\n<p>Therefore, we have an initial-value problem to solve:<\/p>\n<div id=\"fs-id1165043219190\" class=\"equation unnumbered\">[latex]v^{\\prime}(t)=-15, \\, v(0)=88[\/latex].<\/div>\n<p>Integrating, we find that<\/p>\n<div id=\"fs-id1165043395175\" class=\"equation unnumbered\">[latex]v(t)=-15t+C[\/latex].<\/div>\n<p>Since [latex]v(0)=88, \\, C=88[\/latex]. Thus, the velocity function is<\/p>\n<div id=\"fs-id1165042373710\" class=\"equation unnumbered\">[latex]v(t)=-15t+88[\/latex].<\/div>\n<p>To find how long it takes for the car to stop, we need to find the time [latex]t[\/latex] such that the velocity is zero. Solving [latex]-15t+88=0[\/latex], we obtain [latex]t=\\frac{88}{15}[\/latex] sec.<\/li>\n<li>To find how far the car travels during this time, we need to find the position of the car after [latex]\\frac{88}{15}[\/latex] sec. We know the velocity [latex]v(t)[\/latex] is the derivative of the position [latex]s(t)[\/latex]. Consider the initial position to be [latex]s(0)=0[\/latex]. Therefore, we need to solve the initial-value problem\n<div id=\"fs-id1165043380494\" class=\"equation unnumbered\">[latex]s^{\\prime}(t)=-15t+88, \\, s(0)=0[\/latex].<\/div>\n<p>Integrating, we have<\/p>\n<div id=\"fs-id1165043317270\" class=\"equation unnumbered\">[latex]s(t)=-\\frac{15}{2}t^2+88t+C[\/latex].<\/div>\n<p>Since [latex]s(0)=0[\/latex], the constant is [latex]C=0[\/latex]. Therefore, the position function is<\/p>\n<div id=\"fs-id1165043250997\" class=\"equation unnumbered\">[latex]s(t)=-\\frac{15}{2}t^2+88t[\/latex].<\/div>\n<p>After [latex]t=\\frac{88}{15}[\/latex] sec, the position is [latex]s(\\frac{88}{15})\\approx 258.133[\/latex] ft.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042708223\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042708226\" class=\"exercise\">\n<div id=\"fs-id1165042708229\" class=\"textbox\">\n<p id=\"fs-id1165042708231\">Suppose the car is traveling at the rate of 44 ft\/sec. How long does it take for the car to stop? How far will the car travel?<\/p>\n<\/div>\n<div class=\"solution\">\n<div class=\"textbox shaded\">\n<div class=\"solution\">\n<p id=\"fs-id1165042708242\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q923849\">Show Answer<\/span><\/p>\n<div id=\"q923849\" class=\"hidden-answer\" style=\"display: none\">2.93 \\sec, 64.5 ft<\/div>\n<\/div>\n<\/div>\n<div class=\"commentary\"><\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165043396281\">[latex]v(t)=-15t+44[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043396312\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165042323569\">\n<li>If [latex]F[\/latex] is an antiderivative of [latex]f[\/latex], then every antiderivative of [latex]f[\/latex] is of the form [latex]F(x)+C[\/latex] for some constant [latex]C[\/latex].<\/li>\n<li>Solving the initial-value problem\n<div id=\"fs-id1165043259810\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=f(x),y(x_0)=y_0[\/latex]<\/div>\n<p>requires us first to find the set of antiderivatives of [latex]f[\/latex] and then to look for the particular antiderivative that also satisfies the initial condition.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165042709600\" class=\"textbox exercises\">\n<p id=\"fs-id1165042709603\">For the following exercises, show that [latex]F(x)[\/latex] is an antiderivative of [latex]f(x)[\/latex].<\/p>\n<div class=\"exercise\">\n<div id=\"fs-id1165042333423\" class=\"textbox\">\n<p id=\"fs-id1165042333425\"><strong>1.\u00a0<\/strong>[latex]F(x)=5x^3+2x^2+3x+1, \\, f(x)=15x^2+4x+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-id1165042465569\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042465569\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042465569\">[latex]F^{\\prime}(x)=15x^2+4x+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042364150\" class=\"exercise\">\n<div id=\"fs-id1165042364152\" class=\"textbox\">\n<p id=\"fs-id1165042364154\"><strong>2.\u00a0<\/strong>[latex]F(x)=x^2+4x+1, \\, f(x)=2x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043317346\" class=\"exercise\">\n<div id=\"fs-id1165043317348\" class=\"textbox\">\n<p id=\"fs-id1165043317350\"><strong>3.\u00a0<\/strong>[latex]F(x)=x^2e^x, \\, f(x)=e^x(x^2+2x)[\/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-id1165043327399\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043327399\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043327399\">[latex]F^{\\prime}(x)=2xe^x+x^2e^x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042659411\" class=\"exercise\">\n<div id=\"fs-id1165042659413\" class=\"textbox\">\n<p id=\"fs-id1165042659415\"><strong>4.\u00a0<\/strong>[latex]F(x)= \\cos x, \\, f(x)=\u2212 \\sin x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042640878\" class=\"exercise\">\n<div id=\"fs-id1165042640880\" class=\"textbox\">\n<p id=\"fs-id1165042640882\"><strong>5.\u00a0<\/strong>[latex]F(x)=e^x, \\, f(x)=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-id1165042640926\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042640926\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042640926\">[latex]F^{\\prime}(x)=e^x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042638498\">For the following exercises, find the antiderivative of the function.<\/p>\n<div id=\"fs-id1165042638501\" class=\"exercise\">\n<div id=\"fs-id1165042638503\" class=\"textbox\">\n<p id=\"fs-id1165042638505\"><strong>6.\u00a0<\/strong>[latex]f(x)=\\frac{1}{x^2}+x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043390814\" class=\"exercise\">\n<div id=\"fs-id1165043390816\" class=\"textbox\">\n<p id=\"fs-id1165043390818\"><strong>7.\u00a0<\/strong>[latex]f(x)=e^x-3x^2+ \\sin 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-id1165043390859\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043390859\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043390859\">[latex]F(x)=e^x-x^3- \\cos (x)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042708318\" class=\"exercise\">\n<div id=\"fs-id1165042708321\" class=\"textbox\">\n<p><strong>8.\u00a0<\/strong>[latex]f(x)=e^x+3x-x^2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1165042659509\" class=\"textbox\">\n<p id=\"fs-id1165042659512\"><strong>9.\u00a0<\/strong>[latex]f(x)=x-1+4 \\sin (2x)[\/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-id1165042659554\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042659554\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042659554\">[latex]F(x)=\\frac{x^2}{2}-x-2 \\cos (2x)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042465656\">For the following exercises, find the antiderivative [latex]F(x)[\/latex] of each function [latex]f(x)[\/latex].<\/p>\n<div id=\"fs-id1165042465686\" class=\"exercise\">\n<div id=\"fs-id1165042465688\" class=\"textbox\">\n<p id=\"fs-id1165042465690\"><strong>10.\u00a0<\/strong>[latex]f(x)=5x^4+4x^5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043427405\" class=\"exercise\">\n<div id=\"fs-id1165043427407\" class=\"textbox\">\n<p id=\"fs-id1165043427409\"><strong>11.\u00a0<\/strong>[latex]f(x)=x+12x^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-id1165043108238\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043108238\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043108238\">[latex]F(x)=\\frac{1}{2}x^2+4x^3+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043108281\" class=\"exercise\">\n<div id=\"fs-id1165043108283\" class=\"textbox\">\n<p id=\"fs-id1165043108285\"><strong>12.\u00a0<\/strong>[latex]f(x)=\\frac{1}{\\sqrt{x}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042657750\" class=\"exercise\">\n<div id=\"fs-id1165042657752\" class=\"textbox\">\n<p id=\"fs-id1165042657754\"><strong>13.\u00a0<\/strong>[latex]f(x)=(\\sqrt{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-id1165042657787\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042657787\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042657787\">[latex]F(x)=\\frac{2}{5}(\\sqrt{x})^5+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043219223\" class=\"exercise\">\n<div id=\"fs-id1165043219225\" class=\"textbox\">\n<p id=\"fs-id1165043219227\"><strong>14.\u00a0<\/strong>[latex]f(x)=x^{1\/3}+(2x)^{1\/3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042707199\" class=\"exercise\">\n<div id=\"fs-id1165042707201\" class=\"textbox\">\n<p id=\"fs-id1165042707203\"><strong>15.\u00a0<\/strong>[latex]f(x)=\\frac{x^{1\/3}}{x^{2\/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-id1165042707247\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042707247\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042707247\">[latex]F(x)=\\frac{3}{2}x^{2\/3}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043173829\" class=\"exercise\">\n<div id=\"fs-id1165043173831\" class=\"textbox\">\n<p id=\"fs-id1165043173833\"><strong>16.\u00a0<\/strong>[latex]f(x)=2 \\sin (x)+ \\sin (2x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042364505\" class=\"exercise\">\n<div id=\"fs-id1165042364507\" class=\"textbox\">\n<p id=\"fs-id1165042364510\"><strong>17.\u00a0<\/strong>[latex]f(x)=\\sec^2 (x)+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-id1165042364547\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042364547\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042364547\">[latex]F(x)=x+ \\tan (x)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043281579\" class=\"exercise\">\n<div id=\"fs-id1165043281581\" class=\"textbox\">\n<p id=\"fs-id1165043281583\"><strong>18.\u00a0<\/strong>[latex]f(x)= \\sin x \\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043281655\" class=\"exercise\">\n<div id=\"fs-id1165043281657\" class=\"textbox\">\n<p id=\"fs-id1165043281659\"><strong>19.\u00a0<\/strong>[latex]f(x)= \\sin^2 (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-id1165043424715\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043424715\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043424715\">[latex]F(x)=\\frac{1}{3} \\sin^3 (x)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043424756\" class=\"exercise\">\n<div id=\"fs-id1165043424758\" class=\"textbox\">\n<p id=\"fs-id1165043424760\"><strong>20.\u00a0<\/strong>[latex]f(x)=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042660274\" class=\"exercise\">\n<div id=\"fs-id1165042660276\" class=\"textbox\">\n<p id=\"fs-id1165042660278\"><strong>21.\u00a0<\/strong>[latex]f(x)=\\frac{1}{2} \\csc^2 (x)+\\frac{1}{x^2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042660326\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042660326\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042660326\" class=\"hidden-answer\" style=\"display: none\">[latex]F(x)=-\\frac{1}{2} \\cot (x)-\\frac{1}{x}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043173691\" class=\"exercise\">\n<div id=\"fs-id1165043173693\" class=\"textbox\">\n<p id=\"fs-id1165043173696\"><strong>22.\u00a0<\/strong>[latex]f(x)= \\csc x \\cot x+3x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043173769\" class=\"exercise\">\n<div id=\"fs-id1165043173771\" class=\"textbox\">\n<p id=\"fs-id1165043173773\"><strong>23.\u00a0<\/strong>[latex]f(x)=4 \\csc x \\cot x- \\sec x \\tan x[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q93930\">Show Answer<\/span><\/p>\n<div id=\"q93930\" class=\"hidden-answer\" style=\"display: none\">[latex]F(x)=\u2212 \\sec x-4 \\csc x+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043252042\" class=\"exercise\">\n<div id=\"fs-id1165043252044\" class=\"textbox\">\n<p id=\"fs-id1165043252046\"><strong>24.\u00a0<\/strong>[latex]f(x)=8 \\sec x( \\sec x-4 \\tan x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042676267\" class=\"exercise\">\n<div id=\"fs-id1165042676269\" class=\"textbox\">\n<p id=\"fs-id1165042676271\"><strong>25.\u00a0<\/strong>[latex]f(x)=\\frac{1}{2}e^{-4x}+ \\sin 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-id1165042676311\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042676311\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042676311\">[latex]F(x)=-\\frac{1}{8}e^{-4x}- \\cos x+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042681088\">For the following exercises, evaluate the integral.<\/p>\n<div id=\"fs-id1165042681091\" class=\"exercise\">\n<div id=\"fs-id1165042681094\" class=\"textbox\">\n<p id=\"fs-id1165042681096\"><strong>26.\u00a0<\/strong>[latex]\\int (-1) dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042681137\" class=\"exercise\">\n<div id=\"fs-id1165042681139\" class=\"textbox\">\n<p id=\"fs-id1165042681142\"><strong>27.\u00a0<\/strong>[latex]\\int \\sin x dx[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<div class=\"textbox shaded\">\n<div id=\"fs-id1165042681137\" class=\"exercise\">\n<div class=\"solution\">\n<p id=\"fs-id1165042681164\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q436260\">Show Answer<\/span><\/p>\n<div id=\"q436260\" class=\"hidden-answer\" style=\"display: none\">[latex]\u2212 \\cos x+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042681181\" class=\"exercise\">\n<div id=\"fs-id1165042681183\" class=\"textbox\">\n<p id=\"fs-id1165042681185\"><strong>28.\u00a0<\/strong>[latex]\\int (4x+\\sqrt{x}) dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042461202\" class=\"exercise\">\n<div id=\"fs-id1165042461204\" class=\"textbox\">\n<p id=\"fs-id1165042461206\"><strong>29.\u00a0<\/strong>[latex]\\int \\frac{3x^2+2}{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-id1165042461244\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042461244\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042461244\">[latex]3x-\\frac{2}{x}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042374847\" class=\"exercise\">\n<div id=\"fs-id1165042374849\" class=\"textbox\">\n<p id=\"fs-id1165042374851\"><strong>30.\u00a0<\/strong>[latex]\\int (\\sec x \\tan x+4x) dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042374915\" class=\"exercise\">\n<div id=\"fs-id1165042374917\" class=\"textbox\">\n<p id=\"fs-id1165042374920\"><strong>31.\u00a0<\/strong>[latex]\\int (4\\sqrt{x}+\\sqrt[4]{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-id1165042374957\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042374957\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042374957\">[latex]\\frac{8}{3}x^{3\/2}+\\frac{4}{5}x^{5\/4}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042327522\" class=\"exercise\">\n<div id=\"fs-id1165042327524\" class=\"textbox\">\n<p id=\"fs-id1165042327526\"><strong>32.\u00a0<\/strong>[latex]\\int (x^{-1\/3}-x^{2\/3}) dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042700339\" class=\"exercise\">\n<div id=\"fs-id1165042700341\" class=\"textbox\">\n<p id=\"fs-id1165042700343\"><strong>33.\u00a0<\/strong>[latex]\\int \\frac{14x^3+2x+1}{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-id1165042700388\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042700388\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042700388\">[latex]14x-\\frac{2}{x}-\\frac{1}{2x^2}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042700424\" class=\"exercise\">\n<div id=\"fs-id1165042700427\" class=\"textbox\">\n<p id=\"fs-id1165042700429\"><strong>34.\u00a0<\/strong>[latex]\\int (e^x+e^{\u2212x}) dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042668727\">For the following exercises, solve the initial value problem.<\/p>\n<div id=\"fs-id1165042668731\" class=\"exercise\">\n<div id=\"fs-id1165042668733\" class=\"textbox\">\n<p id=\"fs-id1165042668735\"><strong>35.\u00a0<\/strong>[latex]f^{\\prime}(x)=x^{-3}, \\, f(1)=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-id1165042668780\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042668780\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042668780\">[latex]f(x)=-\\frac{1}{2x^2}+\\frac{3}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042668819\" class=\"exercise\">\n<div id=\"fs-id1165042668821\" class=\"textbox\">\n<p id=\"fs-id1165042668823\"><strong>36.\u00a0<\/strong>[latex]f^{\\prime}(x)=\\sqrt{x}+x^2, \\, f(0)=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042315622\" class=\"exercise\">\n<div id=\"fs-id1165042315624\" class=\"textbox\">\n<p id=\"fs-id1165042315626\"><strong>37.\u00a0<\/strong>[latex]f^{\\prime}(x)= \\cos x+ \\sec^2 (x), \\, f(\\frac{\\pi}{4})=2+\\frac{\\sqrt{2}}{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-id1165042684159\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042684159\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042684159\">[latex]f(x)= \\sin x+ \\tan x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042684192\" class=\"exercise\">\n<div id=\"fs-id1165042684194\" class=\"textbox\">\n<p id=\"fs-id1165042684196\"><strong>38.\u00a0<\/strong>[latex]f^{\\prime}(x)=x^3-8x^2+16x+1, \\, f(0)=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042684316\" class=\"exercise\">\n<div id=\"fs-id1165042684318\" class=\"textbox\">\n<p id=\"fs-id1165042684320\"><strong>39.\u00a0<\/strong>[latex]f^{\\prime}(x)=\\frac{2}{x^2}-\\frac{x^2}{2}, \\, f(1)=0[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165043422382\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043422382\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043422382\" class=\"hidden-answer\" style=\"display: none\">[latex]f(x)=-\\frac{1}{6}x^3-\\frac{2}{x}+\\frac{13}{6}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165043422431\">For the following exercises, find two possible functions [latex]f[\/latex] given the second- or third-order derivatives.<\/p>\n<div id=\"fs-id1165043422439\" class=\"exercise\">\n<div id=\"fs-id1165043422441\" class=\"textbox\">\n<p id=\"fs-id1165043422443\"><strong>40.\u00a0<\/strong>[latex]f^{\\prime \\prime}(x)=x^2+2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042326003\" class=\"exercise\">\n<div id=\"fs-id1165042326005\" class=\"textbox\">\n<p id=\"fs-id1165042326007\"><strong>41.\u00a0<\/strong>[latex]f^{\\prime \\prime}(x)=e^{\u2212x}[\/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-id1165042326038\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042326038\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042326038\">Answers may vary; one possible answer is [latex]f(x)=e^{\u2212x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042326066\" class=\"exercise\">\n<div id=\"fs-id1165042326068\" class=\"textbox\">\n<p id=\"fs-id1165042326070\"><strong>42.\u00a0<\/strong>[latex]f^{\\prime \\prime}(x)=1+x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042326139\" class=\"exercise\">\n<div id=\"fs-id1165042326141\" class=\"textbox\">\n<p id=\"fs-id1165042326144\"><strong>43.\u00a0<\/strong>[latex]f^{\\prime \\prime \\prime}(x)= \\cos x[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q671289\">Show Answer<\/span><\/p>\n<div id=\"q671289\" class=\"hidden-answer\" style=\"display: none\">Answers may vary; one possible answer is [latex]f(x)=\u2212 \\sin x[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042667012\" class=\"exercise\">\n<div id=\"fs-id1165042667014\" class=\"textbox\">\n<p id=\"fs-id1165042667017\"><strong>44.\u00a0<\/strong>[latex]f^{\\prime \\prime \\prime}(x)=8e^{-2x}- \\sin x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042667092\" class=\"exercise\">\n<div id=\"fs-id1165042667094\" class=\"textbox\">\n<p id=\"fs-id1165042667097\"><strong>45.\u00a0<\/strong>A car is being driven at a rate of 40 mph when the brakes are applied. The car decelerates at a constant rate of 10 ft\/sec<sup>2<\/sup>. How long before the car stops?<\/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-id1165042667117\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042667117\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042667117\">5.867 sec<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042667126\" class=\"exercise\">\n<div id=\"fs-id1165042667129\" class=\"textbox\">\n<p id=\"fs-id1165042667131\"><strong>46.\u00a0<\/strong>In the preceding problem, calculate how far the car travels in the time it takes to stop.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042667147\" class=\"exercise\">\n<div id=\"fs-id1165042667149\" class=\"textbox\">\n<p id=\"fs-id1165042667151\"><strong>47.\u00a0<\/strong>You are merging onto the freeway, accelerating at a constant rate of 12 ft\/sec<sup>2<\/sup>. How long does it take you to reach merging speed at 60 mph?<\/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-id1165042667171\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042667171\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042667171\">7.333 sec<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042667181\" class=\"exercise\">\n<div id=\"fs-id1165042498516\" class=\"textbox\">\n<p id=\"fs-id1165042498518\"><strong>48.\u00a0<\/strong>Based on the previous problem, how far does the car travel to reach merging speed?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042498534\" class=\"exercise\">\n<div id=\"fs-id1165042498536\" class=\"textbox\">\n<p id=\"fs-id1165042498538\"><strong>49.\u00a0<\/strong>A car company wants to ensure its newest model can stop in 8 sec when traveling at 75 mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.<\/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-id1165042498554\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042498554\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042498554\">13.75 ft\/sec<sup>2<\/sup><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042498566\" class=\"exercise\">\n<div id=\"fs-id1165042498568\" class=\"textbox\">\n<p id=\"fs-id1165042498570\"><strong>50.\u00a0<\/strong>A car company wants to ensure its newest model can stop in less than 450 ft when traveling at 60 mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042498599\">For the following exercises, find the antiderivative of the function, assuming [latex]F(0)=0[\/latex].<\/p>\n<div id=\"fs-id1165042498620\" class=\"exercise\">\n<div id=\"fs-id1165042498622\" class=\"textbox\">\n<p id=\"fs-id1165042498624\"><strong>51. [T]\u00a0<\/strong>[latex]f(x)=x^2+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-id1165042498658\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042498658\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042498658\">[latex]F(x)=\\frac{1}{3}x^3+2x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042498693\" class=\"exercise\">\n<div id=\"fs-id1165042498695\" class=\"textbox\">\n<p id=\"fs-id1165042498697\"><strong>52. [T]\u00a0<\/strong>[latex]f(x)=4x-\\sqrt{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042631784\" class=\"exercise\">\n<div id=\"fs-id1165042631786\" class=\"textbox\">\n<p id=\"fs-id1165042631788\"><strong>53. [T]\u00a0<\/strong>[latex]f(x)= \\sin x+2x[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<div class=\"textbox shaded\">\n<div id=\"fs-id1165042631784\" class=\"exercise\">\n<div class=\"solution\">\n<p id=\"fs-id1165042631822\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q814324\">Show Answer<\/span><\/p>\n<div id=\"q814324\" class=\"hidden-answer\" style=\"display: none\">[latex]F(x)=x^2- \\cos x+1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042631857\" class=\"exercise\">\n<div id=\"fs-id1165042631859\" class=\"textbox\">\n<p id=\"fs-id1165042631861\"><strong>54. [T]\u00a0<\/strong>[latex]f(x)=e^x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042631917\" class=\"exercise\">\n<div id=\"fs-id1165042631920\" class=\"textbox\">\n<p id=\"fs-id1165042631922\"><strong>55. [T]\u00a0<\/strong>[latex]f(x)=\\frac{1}{(x+1)^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-id1165042418084\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042418084\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042418084\">[latex]F(x)=-\\frac{1}{x+1}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042418126\" class=\"exercise\">\n<div id=\"fs-id1165042418128\" class=\"textbox\">\n<p id=\"fs-id1165042418130\"><strong>56. [T]\u00a0<\/strong>[latex]f(x)=e^{-2x}+3x^2[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042418221\">For the following exercises, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false.<\/p>\n<div id=\"fs-id1165042418225\" class=\"exercise\">\n<div id=\"fs-id1165042418227\" class=\"textbox\">\n<p id=\"fs-id1165042418229\"><strong>57.\u00a0<\/strong>If [latex]f(x)[\/latex] is the antiderivative of [latex]v(x)[\/latex], then [latex]2f(x)[\/latex] is the antiderivative of [latex]2v(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-id1165042518929\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042518929\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042518929\">True<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042518934\" class=\"exercise\">\n<div id=\"fs-id1165042518937\" class=\"textbox\">\n<p id=\"fs-id1165042518939\"><strong>58.\u00a0<\/strong>If [latex]f(x)[\/latex] is the antiderivative of [latex]v(x)[\/latex], then [latex]f(2x)[\/latex] is the antiderivative of [latex]v(2x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042519011\" class=\"exercise\">\n<div id=\"fs-id1165042519013\" class=\"textbox\">\n<p id=\"fs-id1165042519016\"><strong>59.\u00a0<\/strong>If [latex]f(x)[\/latex] is the antiderivative of [latex]v(x)[\/latex], then [latex]f(x)+1[\/latex] is the antiderivative of [latex]v(x)+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-id1165042519085\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042519085\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042519085\">False<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042519090\" class=\"exercise\">\n<div id=\"fs-id1165042519092\" class=\"textbox\">\n<p id=\"fs-id1165042519094\"><strong>60.\u00a0<\/strong>If [latex]f(x)[\/latex] is the antiderivative of [latex]v(x)[\/latex], then [latex](f(x))^2[\/latex] is the antiderivative of [latex](v(x))^2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165042617549\" class=\"definition\">\n<dt>antiderivative<\/dt>\n<dd id=\"fs-id1165042617555\">a function [latex]F[\/latex] such that [latex]F^{\\prime}(x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex] is an antiderivative of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042617603\" class=\"definition\">\n<dt>indefinite integral<\/dt>\n<dd id=\"fs-id1165042617608\">the most general antiderivative of [latex]f(x)[\/latex] is the indefinite integral of [latex]f[\/latex]; we use the notation [latex]\\int f(x) dx[\/latex] to denote the indefinite integral of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042617659\" class=\"definition\">\n<dt>initial value problem<\/dt>\n<dd id=\"fs-id1165042617665\">a problem that requires finding a function [latex]y[\/latex] that satisfies the differential equation [latex]\\frac{dy}{dx}=f(x)[\/latex] together with the initial condition [latex]y(x_0)=y_0[\/latex]<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2031\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus I. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/8b89d172-2927-466f-8661-01abc7ccdba4@2.89\">http:\/\/cnx.org\/contents\/8b89d172-2927-466f-8661-01abc7ccdba4@2.89<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/8b89d172-2927-466f-8661-01abc7ccdba4@2.89<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":311,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus I\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/8b89d172-2927-466f-8661-01abc7ccdba4@2.89\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/8b89d172-2927-466f-8661-01abc7ccdba4@2.89\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2031","chapter","type-chapter","status-publish","hentry"],"part":1878,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/chapters\/2031","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/chapters\/2031\/revisions"}],"predecessor-version":[{"id":2770,"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/chapters\/2031\/revisions\/2770"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/parts\/1878"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/chapters\/2031\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/wp\/v2\/media?parent=2031"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=2031"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/wp\/v2\/contributor?post=2031"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/wp\/v2\/license?post=2031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}