e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
356 lines
16 KiB
HTML
356 lines
16 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/calculus/integration-by-parts.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:49:11 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Integration by Parts</title>
|
||
<meta name="description" content="Integration by Parts is a special method of integration that is often useful when two functions are multiplied together, but is also helpful in other ways">
|
||
|
||
<style>
|
||
.intsymb { font: italic 150% Georgia, Arial; position: relative; top: 1px; padding: 0 1px 0 0.2rem;}
|
||
|
||
.intgl {display:inline-block; margin: -4% 0 4% -1%; transform: translateX(20%) translateY(35%);}
|
||
.intgl .to {text-align:center; width:2em; font: 0.8em Verdana; margin: 0 0 -5px 8px;}
|
||
.intgl .symb {font: 180% Georgia;}
|
||
.intgl .symb:before { content: "\222B";}
|
||
.intgl .brack {font: 180% Georgia;}
|
||
.intgl .from {text-align:center; width:2em; font: 0.8em Verdana; overflow:visible; }
|
||
|
||
.sigma {display:inline-block; margin: -4% 0 4% -1%; transform: translateX(20%) translateY(35%);}
|
||
.sigma .to {text-align:center; width:2em; font: 0.8em Verdana; margin: 0 0 -12px 0;}
|
||
.sigma .symb {font: 200% Georgia; transform: translateY(16%); }
|
||
.sigma .symb:before { content: "\03A3";}
|
||
.sigma .from {text-align:center; width:2em; font: 0.8em Verdana; overflow:visible; }
|
||
|
||
</style>
|
||
|
||
|
||
<!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="preload" href="../images/style/font-champ-bold.ttf" as="font" type="font/ttf" crossorigin="">
|
||
<link rel="preload" href="../style4.css" as="style">
|
||
<link rel="preload" href="../main4.js" as="script">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js" defer="defer"></script>
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async="" src="https://www.googletagmanager.com/gtag/js?id=UA-29771508-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag(){dataLayer.push(arguments);}
|
||
gtag('js', new Date());
|
||
gtag('config', 'UA-29771508-1');
|
||
</script>
|
||
</head>
|
||
|
||
<body id="bodybg" class="adv">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo-adv.svg" alt="Math is Fun Advanced"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
|
||
<h1 class="center">Integration by Parts</h1>
|
||
|
||
<p>Integration by Parts is a special method of integration that is often useful when two functions are multiplied together, but is also helpful in other ways.</p>
|
||
<p>You will see plenty of examples soon, but first let us see the rule:</p>
|
||
<p class="center larger"><span class="intsymb">∫</span>u v dx = u<span class="intsymb">∫</span>v dx −<span class="intsymb">∫</span>u' (<span class="intsymb">∫</span>v dx) dx</p>
|
||
<ul>
|
||
<li><b>u</b> is the function u(x)</li>
|
||
<li><b>v</b> is the function v(x)</li>
|
||
<li><b>u'</b> is the <a href="derivatives-rules.html">derivative</a> of the function u(x)</li>
|
||
</ul>
|
||
<p>The rule as a diagram:</p>
|
||
<p class="center"><img src="images/integral-parts-general.svg" alt="integration by parts general" height="181" width="311"></p>
|
||
<p>Let's get straight into an example:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is <span class="intsymb">∫</span>x cos(x) dx ?</h3>
|
||
<p>OK, we have <b>x</b> <i>multiplied by</i> <b>cos(x)</b>, so integration by parts is a good choice.</p>
|
||
<p>First choose which functions for <b>u</b> and <b>v</b>:</p>
|
||
<ul>
|
||
<li>u = x</li>
|
||
<li>v = cos(x)</li>
|
||
</ul>
|
||
<p class="large">So now it is in the format <span class="intsymb"><b>∫</b></span><b>u v dx</b> we can proceed:</p>
|
||
<p>Differentiate <b>u</b>: <span class="larger">u' = x' = 1</span></p>
|
||
<p>Integrate <b>v</b>: <span class="larger"><span class="intsymb">∫</span>v dx = <span class="intsymb">∫</span>cos(x) dx = sin(x)</span> <i>(see <a href="integration-rules.html">Integration Rules</a>)</i></p>
|
||
<p>Now we can put it together:</p>
|
||
<p class="center"><img src="images/integral-parts-x-cosx.svg" alt="integration by parts x cos(x) dx" height="175" width="309"></p>
|
||
<p>Simplify and solve:</p>
|
||
<div class="so">x sin(x) − <span class="intsymb">∫</span>sin(x) dx </div>
|
||
<div class="so">x sin(x) + cos(x) + C</div>
|
||
<p>Done!</p>
|
||
</div>
|
||
<p> </p>
|
||
<p>So we followed these steps:</p>
|
||
<ul>
|
||
<li>Choose u and v</li>
|
||
<li>Differentiate u: u'</li>
|
||
<li>Integrate v: <span class="intsymb">∫</span>v dx</li>
|
||
<li>Put u, u' and<span class="intsymb"> ∫</span>v dx into: <span class="center"><b>u<span class="intsymb">∫</span>v dx −<span class="intsymb">∫</span>u' (<span class="intsymb">∫</span>v dx) dx</b></span></li>
|
||
<li>Simplify and solve</li>
|
||
</ul><br>
|
||
<div class="words">
|
||
<p>In English we can say that <b><span class="intsymb">∫</span>u v dx</b> becomes:</p>
|
||
<p class="center larger">(u integral v) minus integral of (derivative u, integral v)</p>
|
||
</div>
|
||
<p> </p>
|
||
<p>Let's try some more examples:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is<span class="intsymb"> ∫</span>ln(x)/x<sup>2</sup> dx ?</h3>
|
||
<p>First choose u and v:</p>
|
||
<ul>
|
||
<li>u = ln(x)</li>
|
||
<li>v = 1/x<sup>2</sup></li>
|
||
</ul>
|
||
<p>Differentiate u: <span class="larger">ln(x)' = <span class="intbl"><em>1</em><strong>x</strong></span></span></p>
|
||
<p>Integrate v: <span class="larger"><span class="intsymb">∫</span>1/x<sup>2</sup> dx = <span class="intsymb">∫</span>x<sup>-2</sup> dx = −x<sup>-1</sup> = <span class="intbl"><em>−1</em><strong>x</strong></span> </span> <i> (by the <a href="integration-rules.html">power rule</a>)</i></p>
|
||
<p>Now put it together:</p>
|
||
<p class="center"><img src="images/integral-parts-lnx-on-x2.svg" alt="integration by parts ln(x) on x^2" height="191" width="300"></p>
|
||
<p>Simplify:</p>
|
||
<div class="so">−ln(x)/x −<span class="larger"><span class="intsymb"> ∫</span></span>−<span class="larger">1/x<sup>2</sup></span> dx </div>
|
||
<div class="so">−ln(x)/x − 1/x + C</div>
|
||
<div class="so">− <span class="intbl"><em>ln(x) + 1</em><strong>x</strong></span> + C</div>
|
||
<!-- − ln(x)+1/x + C -->
|
||
</div>
|
||
<p> </p><br>
|
||
<p><br></p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is <span class="intsymb">∫</span>ln(x) dx ?</h3>
|
||
<p>But there is only one function! How do we choose u and v ?</p>
|
||
<p>Hey! We can just choose v as being "1":</p>
|
||
<ul>
|
||
<li>u = ln(x)</li>
|
||
<li>v = 1</li>
|
||
</ul>
|
||
<p>Differentiate u: <span class="larger">ln(x)' = 1</span>/x</p>
|
||
<p>Integrate v: <span class="larger"><span class="intsymb">∫</span>1 dx = x </span></p>
|
||
<p>Now put it together:</p>
|
||
<p class="center"><img src="images/integral-parts-lnx.svg" alt="integration by parts ln(x)" height="185" width="303"></p>
|
||
<p>Simplify:</p>
|
||
<div class="so">x ln(x) − <span class="larger"><span class="intsymb">∫</span></span>1<span class="larger"></span> dx </div>
|
||
<div class="so">x ln(x) − x + C</div>
|
||
</div>
|
||
<p> </p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is<span class="intsymb"> ∫</span>e<sup>x</sup> x dx ?</h3>
|
||
<p>Choose u and v:</p>
|
||
<ul>
|
||
<li>u = e<sup>x</sup></li>
|
||
<li>v = x</li>
|
||
</ul>
|
||
<p>Differentiate u: <span class="larger">(e<sup>x</sup>)' = e<sup>x</sup></span></p>
|
||
<p>Integrate v: <span class="larger"> <span class="intsymb">∫</span>x dx = x<sup>2</sup>/2</span></p>
|
||
<p>Now put it together:</p>
|
||
<p class="center"><img src="images/integral-parts-ex-x.svg" alt="integration by parts e^x x" height="188" width="289"></p>
|
||
<p>
|
||
It only got worse!
|
||
</p>
|
||
</div>
|
||
<p>Well, that was a spectacular disaster.</p>
|
||
<p>Maybe we could choose a different u and v?</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: <span class="intsymb">∫</span>e<sup>x</sup> x dx (continued)</h3>
|
||
<p>Choose u and v differently:</p>
|
||
<ul>
|
||
<li>u = x</li>
|
||
<li>v = e<sup>x</sup></li>
|
||
</ul>
|
||
<p>Differentiate u: <span class="larger">(x)' = 1</span></p>
|
||
<p>Integrate v: <span class="larger"><span class="intsymb">∫</span>e<sup>x</sup> dx = e<sup>x</sup></span></p>
|
||
<p>Now put it together:</p>
|
||
<p class="center"><img src="images/integral-parts-x-ex.svg" alt="integration by parts x e^x" height="175" width="285"></p>
|
||
<p>Simplify:</p>
|
||
<div class="so"> x <span class="larger">e<sup>x</sup></span> − <span class="larger">e<sup>x</sup></span> + C </div>
|
||
<div class="so"><span class="larger">e<sup>x</sup></span>(x−1) + C</div>
|
||
</div>
|
||
<p>The moral of the story: Choose <b>u</b> and <b>v</b> carefully!</p>
|
||
<div class="center80">
|
||
<p>Choose a <b>u</b> that gets simpler when you differentiate it and a <b>v</b> that doesn't get any more complicated when you integrate it.</p>
|
||
</div>
|
||
<p>A helpful rule of thumb is <span class="large">I LATE</span>. Choose <b>u</b> based on which of these comes first:</p>
|
||
<ul>
|
||
<li><b>I</b>: <a href="../algebra/trig-inverse-sin-cos-tan.html">Inverse trigonometric functions</a> such as sin<sup>-1</sup>(x), cos<sup>-1</sup>(x), tan<sup>-1</sup>(x)</li>
|
||
<li><b>L</b>: <a href="../sets/function-logarithmic.html">Logarithmic</a> functions such as ln(x), log(x)</li>
|
||
<li><b>A</b>: <a href="../numbers/algebraic-numbers.html">Algebraic</a> functions such as x<sup>2</sup>, x<sup>3</sup></li>
|
||
<li><b>T</b>: <a href="../sine-cosine-tangent.html">Trigonometric functions</a> such as sin(x), cos(x), tan (x)</li>
|
||
<li><b>E</b>: <a href="../sets/function-exponential.html">Exponential functions</a> such as e<sup>x</sup>, 3<sup>x</sup></li>
|
||
</ul><br><p>And here is one last (and tricky) example:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: <span class="intsymb">∫</span>e<sup>x</sup> sin(x) dx</h3>
|
||
<p>Choose u and v:</p>
|
||
<ul>
|
||
<li>u = sin(x)</li>
|
||
<li>v = e<sup>x</sup></li>
|
||
</ul>
|
||
<p>Differentiate u: <span class="larger">sin(x)' = cos(x)</span></p>
|
||
<p>Integrate v: <span class="larger"><span class="intsymb">∫</span>e<sup>x</sup> dx = e<sup>x</sup></span></p>
|
||
<p>Now put it together:</p>
|
||
<div class="so"><span class="intsymb">∫</span>e<sup>x</sup> sin(x) dx = sin(x) e<sup>x</sup> −<span class="intsymb">∫</span>cos(x) e<sup>x</sup> dx</div>
|
||
<p> </p>
|
||
<p>It looks worse, but let us persist! To find <span class="large"><span class="intsymb">∫</span>cos(x) e<sup>x</sup> dx</span> we can use integration by parts <b>again</b>:</p>
|
||
<p>Choose u and v:</p>
|
||
<ul>
|
||
<li>u = cos(x)</li>
|
||
<li>v = e<sup>x</sup></li>
|
||
</ul>
|
||
<p>Differentiate u: <span class="larger">cos(x)' = -sin(x)</span></p>
|
||
<p>Integrate v: <span class="larger"><span class="intsymb">∫</span>e<sup>x</sup> dx = e<sup>x</sup></span></p>
|
||
<p>Now put it together:</p>
|
||
<div class="so"><span class="intsymb">∫</span>e<sup>x</sup> sin(x) dx = sin(x) e<sup>x</sup> − (cos(x) e<sup>x</sup> −<span class="intsymb">∫</span>−sin(x) e<sup>x</sup> dx)</div>
|
||
<p>Simplify:</p>
|
||
<div class="so"><span class="intsymb">∫</span>e<sup>x</sup> sin(x) dx = e<sup>x</sup> sin(x) − e<sup>x</sup> cos(x) −<span class="intsymb">∫</span> e<sup>x</sup> sin(x)dx</div>
|
||
<p>Now we have the <b>same integral on both sides</b> (except one is subtracted) ...</p>
|
||
<p>... so we can bring the right hand integral over to the left and we get:</p>
|
||
<div class="so">2<span class="intsymb">∫</span>e<sup>x</sup> sin(x) dx = e<sup>x</sup> sin(x) − e<sup>x</sup> cos(x)</div>
|
||
<p>Simplify:</p>
|
||
<div class="so"><span class="intsymb">∫</span>e<sup>x</sup> sin(x) dx = ½ e<sup>x</sup> (sin(x) − cos(x)) + C</div>
|
||
</div>
|
||
<p> </p>
|
||
|
||
<h2>Definite Integrals</h2>
|
||
<p>When the integral has an interval like [a, b] we can use either of these:</p><span style="font-size:140%;"></span>
|
||
|
||
|
||
<div class="center larger">
|
||
<div class="intgl">
|
||
<div class="to">b</div>
|
||
<div class="symb"></div>
|
||
<div class="from">a</div>
|
||
</div>u v dx =
|
||
<span style="display:inline-block; transform: scaleY(2);">[</span>
|
||
u<div class="intgl">
|
||
<div class="to"> </div>
|
||
<div class="symb"></div>
|
||
<div class="from"> </div>
|
||
</div>v dx
|
||
|
||
−<div class="intgl">
|
||
<div class="to"> </div>
|
||
<div class="symb"></div>
|
||
<div class="from"> </div>
|
||
</div>u'(<div class="intgl">
|
||
<div class="to"> </div>
|
||
<div class="symb"></div>
|
||
<div class="from"> </div>
|
||
</div>v dx) dx
|
||
|
||
|
||
<span style="display:inline-block; transform: scaleY(2);">]</span>
|
||
<div style="display:inline-block; font: 0.8em Verdana; text-align:center;transform: translateY(30%) translateX(-30%);">
|
||
<div>b</div><br>
|
||
<div>a</div>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
<div class="center larger">
|
||
<div class="intgl">
|
||
<div class="to">b</div>
|
||
<div class="symb"></div>
|
||
<div class="from">a</div>
|
||
</div>u v dx =
|
||
<span style="display:inline-block; transform: scaleY(2);">[</span>
|
||
u<div class="intgl">
|
||
<div class="to"> </div>
|
||
<div class="symb"></div>
|
||
<div class="from"> </div>
|
||
</div>v dx
|
||
<span style="display:inline-block; transform: scaleY(2);">]</span>
|
||
<div style="display:inline-block; font: 0.8em Verdana; text-align:center;transform: translateY(30%) translateX(-30%);">
|
||
<div>b</div><br>
|
||
<div>a</div>
|
||
</div>
|
||
−<div class="intgl">
|
||
<div class="to">b</div>
|
||
<div class="symb"></div>
|
||
<div class="from">a</div>
|
||
</div>u'(<div class="intgl">
|
||
<div class="to"> </div>
|
||
<div class="symb"></div>
|
||
<div class="from"> </div>
|
||
</div>v dx) dx</div>
|
||
|
||
<p>Where u and v are functions of x, and a and b are the limits on x. </p>The second version can help us see the relationship between the left and right integrals.<br>
|
||
<p>See <a href="integration-definite.html">Definite Integrals</a> for more info.</p>
|
||
<span style="font-size:140%;"></span>
|
||
|
||
|
||
<p></p>
|
||
|
||
|
||
<h2>Footnote: Where Did "<span class="center">Integration by Parts</span>" Come From?</h2>
|
||
|
||
<div class="center80">
|
||
<p>It is based on the <a href="derivatives-rules.html">Product Rule for Derivatives</a>:</p>
|
||
<div class="so">(uv)' = uv' + u'v</div>
|
||
<p>Integrate both sides and rearrange:</p>
|
||
<div class="so"><span class="intsymb">∫</span>(uv)' dx = <span class="intsymb">∫</span>uv' dx + <span class="intsymb">∫</span>u'v dx</div>
|
||
<div class="so">uv = <span class="intsymb">∫</span>uv' dx + <span class="intsymb">∫</span>u'v dx</div>
|
||
<div class="so"><span class="intsymb">∫</span>uv' dx = uv − <span class="intsymb">∫</span>u'v dx</div>
|
||
<p>Some people prefer that last form, but I like to replace <b>v' with w</b> and <b>v with<span class="intsymb">∫</span>w dx</b> which makes the left side simpler:</p>
|
||
<div class="so"><span class="intsymb">∫</span>uw dx = u<span class="intsymb">∫</span>w dx − <span class="intsymb">∫</span>u'(<span class="intsymb">∫</span>w dx) dx</div>
|
||
</div>
|
||
<p> </p>
|
||
<div class="questions">6844, 6845, 6846, 6847, 6848, 6849, 6850, 6851, 6852, 6853</div>
|
||
|
||
<div class="related">
|
||
<a href="integration-rules.html">Integration Rules</a>
|
||
<a href="index.html">Calculus Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2022 Rod Pierce</div>
|
||
|
||
</div>
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/calculus/integration-by-parts.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:49:12 GMT -->
|
||
</html> |