{"id":4103,"date":"2022-04-14T18:15:54","date_gmt":"2022-04-14T18:15:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-conservative-vector-fields\/"},"modified":"2022-11-09T16:45:30","modified_gmt":"2022-11-09T16:45:30","slug":"skills-review-for-greens-theorem-divergence-and-curl-and-surface-integrals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-greens-theorem-divergence-and-curl-and-surface-integrals\/","title":{"raw":"Skills Review for Green's Theorem, Divergence and Curl, and Surface Integrals","rendered":"Skills Review for Green&#8217;s Theorem, Divergence and Curl, and Surface Integrals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Apply basic derivative rules<\/li>\r\n \t<li>Apply the chain rule together with the power and product rule<\/li>\r\n \t<li>Find the general antiderivative of a given function<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the section about Green's Theorem, we will learn how Green's Theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. Then,\u00a0 we will learn about divergence and curl, two important operations on vector fields. Lastly, we will explore how to integrate over a surface. Here we will review basic differentiation techniques and basic integration techniques.\r\n<h2>Basic Derivative Rules<\/h2>\r\n<strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-line-integrals\/\" target=\"_blank\" rel=\"noopener\">Module 6, Skills Review for Line Integrals and Conservative Vector Fields<\/a>)<\/em><\/strong>\r\n<h2>The Chain Rule<\/h2>\r\n<strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-line-integrals\/\" target=\"_blank\" rel=\"noopener\">Module 6, Skills Review for Line Integrals and Conservative Vector Fields<\/a>)<\/em><\/strong>\r\n<h2>Apply Basic Integration Techniques<\/h2>\r\n<strong><em>(also in Module 5, Skills Review for Double and Triple Integrals)<\/em><\/strong>\r\n<h2>Indefinite Integrals<\/h2>\r\n<div class=\"textbox shaded\">\r\n<div class=\"title\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\r\n<\/div>\r\n<p id=\"fs-id1165043393369\">Given a function [latex]f[\/latex], the <strong>indefinite integral<\/strong> of [latex]f[\/latex], denoted<\/p>\r\n\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int f(x) dx[\/latex],<\/div>\r\n<div><\/div>\r\n<div><\/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\" style=\"text-align: center;\">[latex]\\displaystyle\\int f(x) dx=F(x)+C[\/latex]<\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><\/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<div id=\"fs-id1165043041347\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Power Rule for Integrals<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165042514785\">For [latex]n \\ne \u22121[\/latex],<\/p>\r\n\r\n<div id=\"fs-id1165043250161\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int x^n dx=\\dfrac{x^{n+1}}{n+1}+C[\/latex]<\/div>\r\n&nbsp;\r\n\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 href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/appendix-b-table-of-derivatives\/\" target=\"_blank\" rel=\"noopener\">Appendix B: Table of Derivatives<\/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]\\displaystyle\\int kdx=\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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<div id=\"fs-id1165043425485\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Properties of Indefinite Integrals<\/h3>\r\n\r\n<hr \/>\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&nbsp;\r\n<p id=\"fs-id1165043393659\"><strong>Sums and Differences<\/strong><\/p>\r\n\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int (f(x) \\pm g(x)) dx=F(x) \\pm G(x)+C[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042328714\"><strong>Constant Multiples<\/strong><\/p>\r\n\r\n<div id=\"fs-id1165042328717\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int kf(x) dx=kF(x)+C[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043248811\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating Indefinite Integrals<\/h3>\r\nEvaluate each of the following indefinite integrals:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\displaystyle\\int (5x^3-7x^2+3x+4) dx[\/latex]<\/li>\r\n \t<li>[latex]\\displaystyle\\int \\frac{x^2+4\\sqrt[3]{x}}{x} dx[\/latex]<\/li>\r\n \t<li>[latex]\\displaystyle\\int \\frac{4}{1+x^2} dx[\/latex]<\/li>\r\n \t<li>[latex]\\displaystyle\\int \\tan x \\cos x dx[\/latex]<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1165042705917\" class=\"exercise\">[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 the properties of indefinite integrals, we can integrate each of the four terms in the integrand separately. We obtain\r\n<div id=\"fs-id1165042552227\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int (5x^3-7x^2+3x+4) dx=\\displaystyle\\int 5x^3 dx-\\displaystyle\\int 7x^2 dx+\\displaystyle\\int 3x dx+\\displaystyle\\int 4 dx[\/latex]<\/div>\r\nFrom the Constant Multiples property of indefinite integrals, each coefficient can be written in front of the integral sign, which gives\r\n<div id=\"fs-id1165043312575\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int 5x^3 dx-\\displaystyle\\int 7x^2 dx+\\displaystyle\\int 3x dx+\\displaystyle\\int 4 dx=5\\displaystyle\\int x^3 dx-7\\displaystyle\\int x^2 dx+3\\displaystyle\\int x dx+4\\displaystyle\\int 1 dx[\/latex]<\/div>\r\nUsing the power rule for integrals, we conclude that\r\n<div id=\"fs-id1165042407363\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\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\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[latex]\\begin{array}{ll} \\displaystyle\\int (x+\\frac{4}{x^{2\/3}}) dx &amp; =\\displaystyle\\int x dx+4\\displaystyle\\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 the properties of indefinite integrals, write the integral as\r\n<div id=\"fs-id1165043348665\" class=\"equation unnumbered\">[latex]4\\displaystyle\\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\" style=\"text-align: center;\">[latex]\\displaystyle\\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\" style=\"text-align: center;\">[latex]\\displaystyle\\int \\tan x \\cos x dx=\\displaystyle\\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 class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nEvaluate [latex]\\displaystyle\\int (4x^3-5x^2+x-7) dx[\/latex]\r\n\r\n[reveal-answer q=\"4078823\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"4078823\"]\r\n\r\nIntegrate each term in the integrand separately, making use of the power rule.\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1165043259694\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043259694\"]\r\n\r\n[latex]x^4-\\frac{5}{3}x^3+\\frac{1}{2}x^2-7x+C[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]210143[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Apply basic derivative rules<\/li>\n<li>Apply the chain rule together with the power and product rule<\/li>\n<li>Find the general antiderivative of a given function<\/li>\n<\/ul>\n<\/div>\n<p>In the section about Green&#8217;s Theorem, we will learn how Green&#8217;s Theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. Then,\u00a0 we will learn about divergence and curl, two important operations on vector fields. Lastly, we will explore how to integrate over a surface. Here we will review basic differentiation techniques and basic integration techniques.<\/p>\n<h2>Basic Derivative Rules<\/h2>\n<p><strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-line-integrals\/\" target=\"_blank\" rel=\"noopener\">Module 6, Skills Review for Line Integrals and Conservative Vector Fields<\/a>)<\/em><\/strong><\/p>\n<h2>The Chain Rule<\/h2>\n<p><strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-line-integrals\/\" target=\"_blank\" rel=\"noopener\">Module 6, Skills Review for Line Integrals and Conservative Vector Fields<\/a>)<\/em><\/strong><\/p>\n<h2>Apply Basic Integration Techniques<\/h2>\n<p><strong><em>(also in Module 5, Skills Review for Double and Triple Integrals)<\/em><\/strong><\/p>\n<h2>Indefinite Integrals<\/h2>\n<div class=\"textbox shaded\">\n<div class=\"title\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<\/div>\n<p id=\"fs-id1165043393369\">Given a function [latex]f[\/latex], the <strong>indefinite integral<\/strong> of [latex]f[\/latex], denoted<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int f(x) dx[\/latex],<\/div>\n<div><\/div>\n<div><\/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\" style=\"text-align: center;\">[latex]\\displaystyle\\int f(x) dx=F(x)+C[\/latex]<\/div>\n<div><\/div>\n<div><\/div>\n<div><\/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<div id=\"fs-id1165043041347\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Power Rule for Integrals<\/h3>\n<hr \/>\n<p id=\"fs-id1165042514785\">For [latex]n \\ne \u22121[\/latex],<\/p>\n<div id=\"fs-id1165043250161\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int x^n dx=\\dfrac{x^{n+1}}{n+1}+C[\/latex]<\/div>\n<p>&nbsp;<\/p>\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 href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/appendix-b-table-of-derivatives\/\" target=\"_blank\" rel=\"noopener\">Appendix B: Table of Derivatives<\/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]\\displaystyle\\int kdx=\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\int \\frac{1}{x\\sqrt{x^2-1}} dx= \\sec^{-1} |x|+C[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165043425485\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Properties of Indefinite Integrals<\/h3>\n<hr \/>\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>&nbsp;<\/p>\n<p id=\"fs-id1165043393659\"><strong>Sums and Differences<\/strong><\/p>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int (f(x) \\pm g(x)) dx=F(x) \\pm G(x)+C[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042328714\"><strong>Constant Multiples<\/strong><\/p>\n<div id=\"fs-id1165042328717\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int kf(x) dx=kF(x)+C[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165043248811\" class=\"textbook exercises\">\n<h3>Example: Evaluating Indefinite Integrals<\/h3>\n<p>Evaluate each of the following indefinite integrals:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\displaystyle\\int (5x^3-7x^2+3x+4) dx[\/latex]<\/li>\n<li>[latex]\\displaystyle\\int \\frac{x^2+4\\sqrt[3]{x}}{x} dx[\/latex]<\/li>\n<li>[latex]\\displaystyle\\int \\frac{4}{1+x^2} dx[\/latex]<\/li>\n<li>[latex]\\displaystyle\\int \\tan x \\cos x dx[\/latex]<\/li>\n<\/ol>\n<div id=\"fs-id1165042705917\" class=\"exercise\">\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 the properties of indefinite integrals, we can integrate each of the four terms in the integrand separately. We obtain\n<div id=\"fs-id1165042552227\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int (5x^3-7x^2+3x+4) dx=\\displaystyle\\int 5x^3 dx-\\displaystyle\\int 7x^2 dx+\\displaystyle\\int 3x dx+\\displaystyle\\int 4 dx[\/latex]<\/div>\n<p>From the Constant Multiples property of indefinite integrals, each coefficient can be written in front of the integral sign, which gives<\/p>\n<div id=\"fs-id1165043312575\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int 5x^3 dx-\\displaystyle\\int 7x^2 dx+\\displaystyle\\int 3x dx+\\displaystyle\\int 4 dx=5\\displaystyle\\int x^3 dx-7\\displaystyle\\int x^2 dx+3\\displaystyle\\int x dx+4\\displaystyle\\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\" style=\"text-align: center;\">[latex]\\displaystyle\\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\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[latex]\\begin{array}{ll} \\displaystyle\\int (x+\\frac{4}{x^{2\/3}}) dx & =\\displaystyle\\int x dx+4\\displaystyle\\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 the properties of indefinite integrals, write the integral as\n<div id=\"fs-id1165043348665\" class=\"equation unnumbered\">[latex]4\\displaystyle\\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\" style=\"text-align: center;\">[latex]\\displaystyle\\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\" style=\"text-align: center;\">[latex]\\displaystyle\\int \\tan x \\cos x dx=\\displaystyle\\int \\sin x dx=\u2212 \\cos x+C[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Evaluate [latex]\\displaystyle\\int (4x^3-5x^2+x-7) dx[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q4078823\">Hint<\/span><\/p>\n<div id=\"q4078823\" class=\"hidden-answer\" style=\"display: none\">\n<p>Integrate each term in the integrand separately, making use of the power rule.<\/p>\n<\/div>\n<\/div>\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>[latex]x^4-\\frac{5}{3}x^3+\\frac{1}{2}x^2-7x+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm210143\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=210143&theme=oea&iframe_resize_id=ohm210143&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-4103\">\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 Volume 1. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" 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