{"id":158,"date":"2019-08-22T18:40:42","date_gmt":"2019-08-22T18:40:42","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=158"},"modified":"2019-12-02T05:41:45","modified_gmt":"2019-12-02T05:41:45","slug":"antiderivatives","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/antiderivatives\/","title":{"rendered":"Antiderivatives"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Antiderivatives &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nLet $f(x)$ be continuous on $[a,b]$.  If $G(x)$ is continuous on\n$[a,b]$ and $G'(x)=f(x)$ for all $x\\in (a,b)$, then $G$ is called an\n<b>antiderivative<\/b> of $f$.\n\n<\/p>\n\n\n\n<p>\nWe can construct antiderivatives by integrating.  The function\n\\[F(x)=\\int^x_a f(t)\\, dt\\]\nis an antiderivative for $f$ since it can be shown that $F(x)$\nconstructed in this way is continuous on $[a,b]$ and $F'(x)=f(x)$ for\nall $x\\in (a,b)$.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Properties<\/h4>\n\n\n\n<p>\n\nLet $F(x)$ be any antiderivative for $f(x)$.\n\n<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> For any constant $C$, $F(x)+C$ is an antiderivative for $f(x)$. <p> <u>Proof:<\/u>  Since $\\displaystyle \\frac{d}{dx}[F(x)]=f(x)$,  \\begin{eqnarray} \\frac{d}{dx}[F(x)+C]&amp;=&amp;\\frac{d}{dx}[F(x)]+\\frac{d}{dx}[C]\\\\ &amp;=&amp;f(x)+0\\\\ &amp;=&amp;f(x) \\end{eqnarray} so $F(x)+C$ is an antiderivative for $f(x)$.  <\/p><\/li><li> Every antiderivative of $f(x)$ can be written in the form  \\[F(x)+C\\] for some $C$.  That is, every two antiderivatives of $f$ differ by at most a constant. <p> <u>Proof:<\/u>  Let $F(x)$ and $G(x)$ be antiderivatives of $f(x)$.  Then $F'(x)=G'(x)=f(x)$, so $F(x)$ and $G(x)$ differ by at most a constant, which requires proof&#8212;it is shown in most calculus texts and is a consequence of the Mean Value Theorem. <\/p><\/li><\/ul>\n\n\n\n<p>\nThe process of finding antiderivatives is called\n<b>antidifferentiation<\/b> or <b>integration<\/b>:\n\\[\n\\begin{array}{l@{\\qquad}l@{\\qquad}l}\n\t\\displaystyle\\frac{d}{dx}[F(x)]=f(x) &amp; \\Longleftrightarrow &amp;\n\t\\displaystyle\\int f(x)\\, dx=F(x)+C.\\\\\n\t\\displaystyle\\frac{d}{dx}[g(x)]=g'(x) &amp; \\Longleftrightarrow &amp;\n\t\\displaystyle\\int g'(x)\\, dx=g(x)+C. \n\\end{array}\n\\]\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Properties of the Indefinite Integral<\/h4>\n\n\n\n<ul class=\"wp-block-list\"><li> $\\displaystyle \\frac{d}{dx}\\left[\\int\\! f(x)\\, dx\\right]=f(x)$.\n\t\t<p>\n\t\t<u>Proof:<\/u>\n\t\t<\/p><p>\n\t\tLet $\\displaystyle\\int f(x)\\, dx=F(x)$, where $F(x)$ is an antiderivative\n\t\tof $f$.  Then\n\t\t\\begin{eqnarray*}\n\t\t\t\\frac{d}{dx}\\left[\\int f(x)\\, dx\\right]&amp;=&amp;\\frac{d}{dx}F(x)\\\\\n\t\t\t&amp;=&amp;f(x).\n\t\t\\end{eqnarray*}\n\n\t<\/p><\/li><li> <i>(Linearity)<\/i> $\\displaystyle \\int [\\alpha f(x)+\\beta\n\tg(x)]\\, dx=\\alpha \\int\\! f(x)\\, dx+\\beta \\int\\! g(x)\\, dx$.\n\t\t<p>\n\t\t<u>Proof:<\/u>\n\t\t<\/p><p>\n\t\tWe need only show that $\\displaystyle \\alpha\\!\\int\\! f(x)\\, dx+\\beta\\!\\int\\!\n\t\tg(x)\\, dx$ is an antiderivative of $\\displaystyle\\int\\! [\\alpha f(x)+\\beta\n\t\tg(x)]\\, dx$:\n\t\t\\begin{eqnarray*}\n\t\t\t\\frac{d}{dx}\\left[\\alpha \\int f(x)\\, dx+\\beta \\int g(x)\\, dx\\right]&amp;=&amp;\n\t\t\t\\alpha \\frac{d}{dx}\\left[\\int f(x)\\, dx\\right]+\\beta\n\t\t\t\\frac{d}{dx}\\left[\\int g(x)\\, dx\\right]\\\\\n\t\t\t&amp;=&amp;\\alpha f(x)+\\beta g(x).\n\t\t\\end{eqnarray*}\n<\/p><\/li><\/ul>\n\n\n\n<h6 class=\"wp-block-heading\">Examples<\/h6>\n\n\n\n<ol class=\"wp-block-list\"><li> Every antiderivative of $x^2$ has the form $\\displaystyle\n\t\t\\frac{x^3}{3}+C$, since $\\displaystyle\n\t\t\\frac{d}{dx}\\left[\\frac{x^3}{3}\\right]=x^2$.\n\n\t<\/li><li> $\\displaystyle \\frac{d}{dx}\\left[\\int\\! x^5\\, dx\\right]=x^5$.\n<\/li><\/ol>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<h4>Key Concepts<\/h4><h4>\n<\/h4><\/center>\n\n\n\n<p>\n\nIf $G(x)$ is continuous on $[a,b]$ and $G'(x) = f(x)$ for all $x\\in (a,b)$, \nthen $G$ is called an antiderivative of $f$.\n<\/p>\n\n\n\n<p>\nWe can construct antiderivatives by integrating. The function $F(x) = \n\\displaystyle\\int^x_a f(t)\\, dt$ is an antiderivative for $f$. In fact, every antiderivative of $f(x)$ can be written in the form $F(x)+C$, for some $C$.<\/p>\n\n\n\n<p>\n\\[\n\\begin{array}{l@{\\qquad}l@{\\qquad}l}\n\t\\displaystyle\\frac{d}{dx}[F(x)]=f(x) &amp; \\Longleftrightarrow &amp;\n\t\\displaystyle\\int f(x)\\, dx=F(x)+C.\\\\\n\t\\displaystyle\\frac{d}{dx}[g(x)]=g'(x) &amp; \\Longleftrightarrow &amp;\n\t\\displaystyle\\int g'(x)\\, dx=g(x)+C. \n\\end{array}\n\\]\t\n\n<!------------------------>\n\n\n<br>\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ2610\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Antiderivatives &#8211; HMC Calculus Tutorial Let $f(x)$ be continuous on $[a,b]$. If $G(x)$ is continuous on $[a,b]$ and $G'(x)=f(x)$ for all $x\\in (a,b)$, then $G$ is called an antiderivative of $f$. We can construct antiderivatives by integrating. The function \\[F(x)=\\int^x_a f(t)\\, dt\\] is an antiderivative for $f$ since it can be shown that $F(x)$ constructed&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-158","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/158","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=158"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/158\/revisions"}],"predecessor-version":[{"id":1077,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/158\/revisions\/1077"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=158"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=158"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}