{"id":443,"date":"2021-02-04T02:07:43","date_gmt":"2021-02-04T02:07:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=443"},"modified":"2022-03-16T05:58:48","modified_gmt":"2022-03-16T05:58:48","slug":"the-reverse-of-differentiation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/the-reverse-of-differentiation\/","title":{"raw":"Finding the Antiderivative","rendered":"Finding the Antiderivative"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/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<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165043404679\" class=\"bc-section section\">\r\n\r\nWe 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>).\r\n<h2>The Reverse of Differentiation<\/h2>\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 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-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\" style=\"text-align: center;\">[latex]F^{\\prime}(x)=f(x)[\/latex]<\/div>\r\n&nbsp;\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 shaded\">\r\n<h3 style=\"text-align: center;\">General Form of an Antiderivative<\/h3>\r\n\r\n<hr \/>\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=\"textbook exercises\">\r\n<h3>Example: Finding Antiderivatives<\/h3>\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)=\\dfrac{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[reveal-answer q=\"46129\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"46129\"]\r\n\r\na. Because\r\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(x^3)=3x^2[\/latex]<\/p>\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<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\ln x)=\\dfrac{1}{x}[\/latex]<\/p>\r\nFor [latex]x&lt;0, \\, f(x)=\\ln (\u2212x)[\/latex] and\r\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\ln (\u2212x))=-\\dfrac{1}{\u2212x}=\\dfrac{1}{x}[\/latex]<\/p>\r\nTherefore,\r\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\ln |x|)=\\dfrac{1}{x}[\/latex]<\/p>\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<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex],<\/p>\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<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(e^x)=e^x[\/latex],<\/p>\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].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Finding Antiderivatives.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/j81IZAEfwhI?controls=0&amp;start=47&amp;end=158&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.10Antiderivatives47to158_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.10 Antiderivatives\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1165043353933\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind all antiderivatives of [latex]f(x)= \\sin x[\/latex].\r\n\r\n[reveal-answer q=\"8800299\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"8800299\"]\r\n\r\nWhat function has a derivative of [latex] \\sin x[\/latex]?\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"314667\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"314667\"]\r\n\r\n[latex]\u2212\\cos x+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]5318[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Indefinite Integrals<\/h2>\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\" style=\"text-align: center;\">[latex]\\displaystyle\\int f(x) dx=F(x)+C[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042959838\">The symbol [latex]\\displaystyle\\int [\/latex] is called an <em>integral sign<\/em>, and [latex]\\displaystyle\\int f(x) dx[\/latex] is called the indefinite integral of [latex]f[\/latex].<\/p>\r\n\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<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\" style=\"text-align: center;\">[latex]\\displaystyle\\int 2x dx=x^2+C[\/latex]<\/div>\r\n&nbsp;\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]. Figure 1 shows a graph of this family of antiderivatives.<\/p>\r\n\r\n[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\" \/> Figure 1. 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]For 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\" style=\"text-align: center;\">[latex]\\displaystyle\\int x^n dx=\\dfrac{x^{n+1}}{n+1}+C[\/latex],<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165043015098\">which comes directly from<\/p>\r\n\r\n<div id=\"fs-id1165043036022\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}\\left(\\dfrac{x^{n+1}}{n+1}\\right)=(n+1)\\dfrac{x^n}{n+1}=x^n[\/latex]<\/div>\r\n&nbsp;\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 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\/calculus1\/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<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\" style=\"text-align: center;\">[latex]\\displaystyle\\int f(x) dx=F(x)+C[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\displaystyle\\int f(x) dx=F(x)+C[\/latex]<\/div>\r\n&nbsp;\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=\"textbook exercises\">\r\n<h3>Example: Verifying an Indefinite Integral<\/h3>\r\n<p id=\"fs-id1165043428255\">Each of the following statements is of the form [latex]\\displaystyle\\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]\\displaystyle\\int (x+e^x) dx=\\dfrac{x^2}{2}+e^x+C[\/latex]<\/li>\r\n \t<li>[latex]\\displaystyle\\int xe^xdx=xe^x-e^x+C[\/latex]<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1165043393826\" class=\"exercise\">[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\" style=\"text-align: center;\">[latex]\\frac{d}{dx}\\left(\\frac{x^2}{2}+e^x+C\\right)=x+e^x[\/latex],<\/div>\r\nthe statement\r\n<div id=\"fs-id1165042319135\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\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\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[latex]\\displaystyle\\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\" style=\"text-align: center;\">[latex]\\frac{d}{dx}\\left(\\frac{x^2e^x}{2}\\right)=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 id=\"fs-id1165043078178\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nVerify that [latex]\\displaystyle\\int x \\cos x dx=x \\sin x+ \\cos x+C[\/latex].\r\n\r\n[reveal-answer q=\"1770433\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"1770433\"]\r\n\r\nCalculate [latex]\\frac{d}{dx}(x \\sin x+ \\cos x+C)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1165043257533\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043257533\"]\r\n\r\n[latex]\\frac{d}{dx}(x \\sin x+ \\cos x+C)= \\sin x+x \\cos x- \\sin x=x \\cos x[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165042318564\">Earlier, 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 the last example. 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\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(F(x)+G(x))=F^{\\prime}(x)+G^{\\prime}(x)=f(x)+g(x)[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\displaystyle\\int (f(x)+g(x)) dx=F(x)+G(x)+C[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165043174082\">Similarly,<\/p>\r\n\r\n<div id=\"fs-id1165043174085\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int (f(x)-g(x)) dx=F(x)-G(x)+C[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(kf(x))=k\\frac{d}{dx}F(x)=kF^{\\prime}(x)[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\displaystyle\\int kf(x) dx=kF(x)+C[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165043425482\">These properties are summarized next.<\/p>\r\n\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\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 the previous example). 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=\"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\nWatch the following video to see the worked solution to Example: Evaluating Indefinite Integrals.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/j81IZAEfwhI?controls=0&amp;start=409&amp;end=641&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.10Antiderivatives409to641_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.10 Antiderivatives\" here (opens in new window)<\/a>.[\/hidden-answer]\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]210327[\/ohm_question]\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>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<\/ul>\n<\/div>\n<div id=\"fs-id1165043404679\" class=\"bc-section section\">\n<p>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<h2>The Reverse of Differentiation<\/h2>\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 shaded\">\n<div class=\"title\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\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\" style=\"text-align: center;\">[latex]F^{\\prime}(x)=f(x)[\/latex]<\/div>\n<p>&nbsp;<\/p>\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 shaded\">\n<h3 style=\"text-align: center;\">General Form of an Antiderivative<\/h3>\n<hr \/>\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=\"textbook exercises\">\n<h3>Example: Finding Antiderivatives<\/h3>\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)=\\dfrac{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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q46129\">Show Solution<\/span><\/p>\n<div id=\"q46129\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. Because<\/p>\n<p style=\"text-align: center;\">[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 style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\ln x)=\\dfrac{1}{x}[\/latex]<\/p>\n<p>For [latex]x<0, \\, f(x)=\\ln (\u2212x)[\/latex] and\n\n\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\ln (\u2212x))=-\\dfrac{1}{\u2212x}=\\dfrac{1}{x}[\/latex]<\/p>\n<p>Therefore,<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\ln |x|)=\\dfrac{1}{x}[\/latex]<\/p>\n<p>Thus, [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].<\/p>\n<p>c. We have<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex],<\/p>\n<p>so [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].<\/p>\n<p>d. Since<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(e^x)=e^x[\/latex],<\/p>\n<p>then [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].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Finding Antiderivatives.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/j81IZAEfwhI?controls=0&amp;start=47&amp;end=158&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.10Antiderivatives47to158_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.10 Antiderivatives&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1165043353933\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find all antiderivatives of [latex]f(x)= \\sin x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q8800299\">Hint<\/span><\/p>\n<div id=\"q8800299\" class=\"hidden-answer\" style=\"display: none\">\n<p>What function has a derivative of [latex]\\sin x[\/latex]?<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q314667\">Show Solution<\/span><\/p>\n<div id=\"q314667\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\u2212\\cos x+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=\"ohm5318\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5318&theme=oea&iframe_resize_id=ohm5318&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Indefinite Integrals<\/h2>\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\" style=\"text-align: center;\">[latex]\\displaystyle\\int f(x) dx=F(x)+C[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042959838\">The symbol [latex]\\displaystyle\\int[\/latex] is called an <em>integral sign<\/em>, and [latex]\\displaystyle\\int f(x) dx[\/latex] is called the indefinite integral of [latex]f[\/latex].<\/p>\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<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\" style=\"text-align: center;\">[latex]\\displaystyle\\int 2x dx=x^2+C[\/latex]<\/div>\n<p>&nbsp;<\/p>\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]. Figure 1 shows a graph of this family of antiderivatives.<\/p>\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\">Figure 1. 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<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\" style=\"text-align: center;\">[latex]\\displaystyle\\int x^n dx=\\dfrac{x^{n+1}}{n+1}+C[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043015098\">which comes directly from<\/p>\n<div id=\"fs-id1165043036022\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}\\left(\\dfrac{x^{n+1}}{n+1}\\right)=(n+1)\\dfrac{x^n}{n+1}=x^n[\/latex]<\/div>\n<p>&nbsp;<\/p>\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 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\/calculus1\/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<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\" style=\"text-align: center;\">[latex]\\displaystyle\\int f(x) dx=F(x)+C[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]\\displaystyle\\int f(x) dx=F(x)+C[\/latex]<\/div>\n<p>&nbsp;<\/p>\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=\"textbook exercises\">\n<h3>Example: Verifying an Indefinite Integral<\/h3>\n<p id=\"fs-id1165043428255\">Each of the following statements is of the form [latex]\\displaystyle\\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]\\displaystyle\\int (x+e^x) dx=\\dfrac{x^2}{2}+e^x+C[\/latex]<\/li>\n<li>[latex]\\displaystyle\\int xe^xdx=xe^x-e^x+C[\/latex]<\/li>\n<\/ol>\n<div id=\"fs-id1165043393826\" class=\"exercise\">\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\" style=\"text-align: center;\">[latex]\\frac{d}{dx}\\left(\\frac{x^2}{2}+e^x+C\\right)=x+e^x[\/latex],<\/div>\n<p>the statement<\/p>\n<div id=\"fs-id1165042319135\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\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\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[latex]\\displaystyle\\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\" style=\"text-align: center;\">[latex]\\frac{d}{dx}\\left(\\frac{x^2e^x}{2}\\right)=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 id=\"fs-id1165043078178\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Verify that [latex]\\displaystyle\\int x \\cos x dx=x \\sin x+ \\cos x+C[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q1770433\">Hint<\/span><\/p>\n<div id=\"q1770433\" class=\"hidden-answer\" style=\"display: none\">\n<p>Calculate [latex]\\frac{d}{dx}(x \\sin x+ \\cos x+C)[\/latex].<\/p>\n<\/div>\n<\/div>\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>[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>\n<\/div>\n<p id=\"fs-id1165042318564\">Earlier, 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 the last example. 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\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(F(x)+G(x))=F^{\\prime}(x)+G^{\\prime}(x)=f(x)+g(x)[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]\\displaystyle\\int (f(x)+g(x)) dx=F(x)+G(x)+C[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043174082\">Similarly,<\/p>\n<div id=\"fs-id1165043174085\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int (f(x)-g(x)) dx=F(x)-G(x)+C[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(kf(x))=k\\frac{d}{dx}F(x)=kF^{\\prime}(x)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043222034\">for any real number [latex]k[\/latex], we conclude that<\/p>\n<div id=\"fs-id1165042383898\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int kf(x) dx=kF(x)+C[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043425482\">These properties are summarized next.<\/p>\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<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 the previous example). 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=\"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<p>Watch the following video to see the worked solution to Example: Evaluating Indefinite Integrals.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/j81IZAEfwhI?controls=0&amp;start=409&amp;end=641&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.10Antiderivatives409to641_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.10 Antiderivatives&#8221; here (opens in new window)<\/a>.<\/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=\"ohm210327\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=210327&theme=oea&iframe_resize_id=ohm210327&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-443\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>4.10 Antiderivatives. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/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>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/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":17533,"menu_order":37,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"4.10 Antiderivatives\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-443","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/443","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":19,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/443\/revisions"}],"predecessor-version":[{"id":4857,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/443\/revisions\/4857"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/48"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/443\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=443"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=443"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=443"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=443"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}