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<title>Integration by Substitution</title>
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<h1 class="center">Integration by Substitution</h1>
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<p>"Integration by Substitution" (also called "u-Substitution" or "The Reverse Chain Rule") is a method to find an <a href="integration-introduction.html">integral</a>, but only when it can be set up in a special way.</p>
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<p>The first and most vital step is to be able to write our integral in this form:</p>
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<p class="center"><img src="images/integral-subs-1.svg" alt="integration by substitution general"><br>
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Note that we have <b>g(x)</b> and its <a href="derivatives-introduction.html">derivative</a> <b>g'(x)</b></p>
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<p>Like in this example:</p>
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<p class="center">
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<img src="images/integral-subs-cos-x2-2x-a.svg" alt="integration by substitution cos(x^2) 2x dx"><br>
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Here <b>f=cos</b>, and we have <b>g=x<sup>2</sup></b> and its derivative <b>2x</b><br>
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This integral is good to go!</p>
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<p>When our integral is set up like that, we can do <b>this substitution</b>:</p>
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<p class="center"><img src="images/integral-subs-2.svg" alt="integration by substitution general"></p>
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<p>Then we can <b>integrate f(u)</b>, and finish by <b>putting g(x) back as u</b>.</p>
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<p>Like this:</p>
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<div class="example">
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<h3>Example: <span style="font-size:150%;">∫</span>cos(x<sup>2</sup>) 2x dx</h3>
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<p>We know (from above) that it is in the right form to do the substitution:</p>
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<p class="center"><img src="images/integral-subs-3.svg" alt="integration by substitution cos(x^2) 2x dx"></p>
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<p>Now integrate:</p>
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<p class="center larger"><span style="font-size:150%;">∫</span>cos(u) du = sin(u) + C</p>
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<p>And finally put <b>u=x<sup>2</sup></b> back again:</p>
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<p class="center larger">sin(x<sup>2</sup>) + C</p>
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</div>
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<p>So <span style="font-size:150%;"><b>∫</b></span><b>cos(x<sup>2</sup>) 2x dx = sin(x<sup>2</sup>) + C</b></p>
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<p>That worked out really nicely! (Well, I knew it would.)</p>
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<p>But this method only works on <i>some</i> integrals of course, and it may need rearranging:</p>
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<div class="example">
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<h3>Example: <span style="font-size:150%;">∫</span>cos(x<sup>2</sup>) 6x dx</h3>
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<p>Oh no! It is <b>6x</b>, not <b>2x</b> like before. Our perfect setup is gone.</p>
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<p>Never fear! Just rearrange the integral like this:</p>
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<p class="center larger"><span style="font-size:150%;">∫</span>cos(x<sup>2</sup>) 6x dx = 3<span style="font-size:150%;">∫</span>cos(x<sup>2</sup>) 2x dx</p>
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<p>(We can pull constant multipliers outside the integration, see <a href="integration-rules.html">Rules of Integration</a>.)</p>
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<p>Then go ahead as before:</p>
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<p class="center larger">3<span style="font-size:150%;">∫</span>cos(u) du = 3 sin(u) + C</p>
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<p>Now put <b>u=x<sup>2</sup></b> back again:</p>
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<p class="center larger">3 sin(x<sup>2</sup>) + C</p>
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<p>Done!</p>
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</div>
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<p>Now let's try a slightly harder example:</p>
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<div class="example">
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<h3>Example: <span style="font-size:150%;">∫</span>x/(x<sup>2</sup>+1) dx</h3>
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<p>Let me see ... the derivative of x<sup>2</sup>+1 is 2x ... so how about we rearrange it like this:</p>
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<p class="center larger"><span style="font-size:150%;">∫</span>x/(x<sup>2</sup>+1) dx = ½<span style="font-size:150%;">∫</span>2x/(x<sup>2</sup>+1) dx</p>
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<p>Then we have:</p>
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<p class="center larger"><img src="images/integral-subs-4.svg" alt="integration by substitution 2x/(x^2+1)"></p>
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<p>Then integrate:</p>
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<p class="center larger">½<span style="font-size:150%;">∫</span>1/u du = ½ ln<b>|</b>u<b>|</b> + C</p>
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<p>Now put <b>u=x<sup>2</sup>+1</b> back again:</p>
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<p class="center larger">½ ln(x<sup>2</sup>+1) + C</p>
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</div>
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<p>And how about this one:</p>
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<div class="example">
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<h3>Example: <span style="font-size:150%;">∫</span>(x+1)<sup>3</sup> dx</h3>
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<p>Let me see ... the derivative of x+1 is ... well it is simply 1.</p>
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<p>So we can have this:</p>
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<p class="center larger"><span style="font-size:150%;">∫</span>(x+1)<sup>3</sup> dx = <span style="font-size:150%;">∫</span>(x+1)<sup>3</sup> · 1 dx</p>
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<p>Then we have:</p>
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<p class="center"><img src="images/integral-subs-5.svg" alt="integration by substitution (x+1)^3"></p>
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<p>Then integrate:</p>
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<p class="center larger"><span style="font-size:150%;">∫</span>u<sup>3</sup> du = <span class="intbl"><em>u<sup>4</sup></em><strong>4</strong></span> + C</p>
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<p>Now put <b>u=x+1</b> back again:</p>
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<p class="center larger"><span class="intbl"><em>(x+1)<sup>4</sup></em><strong>4</strong></span> + C</p>
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</div>
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<p>We can take that idea further like this:</p>
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<div class="example">
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<h3>Example: <span style="font-size:150%;">∫</span>(5x+2)<sup>7</sup> dx</h3>
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<p>If it was in THIS form we could do it:</p>
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<p class="center larger"><span style="font-size:150%;">∫</span>(5x+2)<sup>7</sup> <b>5</b> dx</p>
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<p>So let's make it so by doing this:</p>
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<p class="center larger"><span class="intbl"><em>1</em><strong>5</strong></span> <span style="font-size:150%;">∫</span>(5x+2)<sup>7</sup> <b>5</b> dx</p>
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<p>The <span class="intbl"><em>1</em><strong>5</strong></span> and 5 cancel out so all is fine.</p>
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<p>And now we can have <b>u=5x+2</b></p>
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<p class="center"><img src="images/integral-subs-6.svg" alt="integration by substitution (x+1)^3"></p>
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<p>And then integrate:</p>
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<p class="center larger"><span class="intbl"><em>1</em><strong>5</strong></span> <span style="font-size:150%;">∫</span>u<sup>7</sup> du = <span class="intbl"><em>1</em><strong>5</strong></span> <span class="intbl"><em>u<sup>8</sup></em><strong>8</strong></span> + C</p>
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<p>Now put <b>u=5x+2</b> back again, and simplify:</p>
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<p class="center larger"><span class="intbl"><em>(5x+2)<sup>8</sup></em><strong>40</strong></span> + C</p>
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</div>
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Now get some practice, OK?
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<h2>In Summary</h2>
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<div class="dotpoint"> When we can put an integral in this form:
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<p class="center"><img src="images/integral-subs-1.svg" alt="integration by substitution general"></p>
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</div>
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<div class="dotpoint"> Then we can set <b>u=g(x)</b> and integrate <span class="center larger"><span style="font-size:150%;">∫</span>f(u) du</span></div>
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<div class="dotpoint"> And finish up by re-inserting <b>g(x)</b> where <b>u</b> is.
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</div>
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<p> </p>
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<div class="questions">
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<script>getQ(6854, 6855, 6856, 6857, 6858, 6859, 6860, 6861, 6862, 6863);</script>
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<div class="related">
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<a href="integration-introduction.html">Introduction to Integration</a>
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<a href="index.html">Calculus Index</a>
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