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id="main-article"> <p><img alt="" src="../../files/integration/by-substitution/image-1.jpg" style="float: left; width: 100px; height: 100px;">Integration by substitution or U-substitution is a method that will help you integrate many different functions. By changing the variable of the integrand, we can make an apparently difficult problem into a much simpler one. The challenge is recognising <strong>when</strong> it is a useful method and <strong>what </strong>the substitution is (it is very rare that this will be suggested to you). On this page, we'll concentrate on doing that. In order to master the techniques, it is vital that you get plenty of practice!</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p>On this page, you should learn about</p> <ul> <li>integration by substitution</li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <p>The first videos are fairly simple examples which use Integration by substitution. In the videos, we will look at the method and how to set it up, but will also concentrate on<strong> how we recognise</strong> that it is an integration by substitution question.</p> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Simple Example 1</p> </div> </div> <div class="panel-body"> <p><span class="math-tex">\(\int { \frac { cosx }{ { \left( sinx-1 \right) }^{ 2 } } dx } \)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/230425888"></iframe></div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Simple Example 2</p> </div> </div> <div class="panel-body"> <div> <div> <p><span class="math-tex">\(\int { 2{ e }^{ x }cos({ e }^{ x })dx } \)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/230425965"></iframe></div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Simple Example 3</p> </div> </div> <div class="panel-body"> <div> <div> <p><span class="math-tex">\(\int { \frac { arctanx }{ x^{ 2 }+1 } } dx\)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/230557476"></iframe></div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Definite Integral</p> </div> </div> <div class="panel-body"> <div> <div> <p>When we are required to evaluate definite integrals then care is required dealing with the limits of integration. It is important to change the limits of integration according to the substitution that you use. The video below shows you how to do that</p> <p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> <span class="math-tex">\(\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 2 } }{ { cos }^{ 3 }xsinxdx } \)</span></p> <p style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="360" mozallowfullscreen="" src="https://player.vimeo.com/video/230727642" webkitallowfullscreen="" width="640"></iframe> </p> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Difficult Example 1 </p> </div> </div> <div class="panel-body"> <div> <p><span class="math-tex">\(\int { { sin }^{ 3 }xdx } \)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/230557521"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Difficult Example 2</p> </div> </div> <div class="panel-body"> <div> <div> <p><span class="math-tex">\(\int { tan^{ 3 }xdx } \)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/230560114"></iframe></div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Difficult Example 3</p> </div> </div> <div class="panel-body"> <div> <div> <p>It is really not obvious at first glance how Integration by substitution helps with the following question. The video concentrates on how you might go through the decision making to decide to use this method. Interestingly, this question can also be solved using Integration by parts. A challenge for you to try for yourself!</p> <p><span class="math-tex">\(\int { x\sqrt { x+1 } dx } \)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/230665352"></iframe></div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-expandable panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Difficult Example 4</p> </div> </div> <div class="panel-body"> <div> <p>In the following example, the substitution has been suggested for us. It would appear to make it an easier type of question. However, it is not! Very careful manipulation is required to get to the correct result.</p> <p>Find <span class="math-tex">\(\int { \frac { x^{ 2 } }{ \sqrt { 36-x^ 2 } } dx } \)</span> using the substitution <span class="math-tex">\( x=sin\theta\)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/230670024"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/integration/by-substitution/revision-notes_integration-by-substitution.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/integration/by-substitution/revision-notes_integration-by-substitution.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>The challenge when you start doing Integration by substitution questions is choosing what the correct substitution is. Remember, you are trying to find a function f(x) within the integrand such that the differential of this, f'(x) is also present in the integrand</p> <p style="text-align: center;"><span class="math-tex">\(\int { g'(f(x)) \ \cdot f'(x)\quad \ dx } \)</span></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#13a05d69"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="13a05d69"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Integration by substitution 1 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-167-634" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The following integral can be performed by using integration by substitution (or U-substitution). Which is the correct substitution?</p><p><span class="math-tex">\(\int { 2x{ e }^{ x^2 }dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> Integration by substitution will not work</label></p><p><label class="radio"><input type="radio"> u = 2x</label></p><p><label class="radio"><input class="c" type="radio"> u = x<sup>2</sup></label></p><p><label class="radio"><input type="radio"> u = <span class="math-tex">\(e ^{ x^2 }\)</span></label></p></div><div class="q-explanation"><p>If u = x<sup>2</sup></p><p>then <span class="math-tex">\(\frac { du }{ dx } =2x\)</span></p><p>Integrand becomes <span class="math-tex">\(\int { { e }^{ u }du } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following integral can be performed by using integration by substitution (or U-substitution). Which is the correct substitution?</p><p><span class="math-tex">\(\int { sin^2x \ cosx \ dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> u = sinx</label></p><p><label class="radio"><input type="radio"> u = x<sup>2</sup></label></p><p><label class="radio"><input type="radio"> u = cosx</label></p><p><label class="radio"><input type="radio"> Integration by substitution will not work</label></p></div><div class="q-explanation"><p>If u = sinx</p><p>then <span class="math-tex">\(\frac { du }{ dx } =cosx\)</span></p><p>Integrand becomes <span class="math-tex">\(\int { { u }^{ 2 }du } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following integral can be performed by using integration by substitution (or U-substitution). Which is the correct substitution?</p><p><span class="math-tex">\(\int { \frac { 3 }{ x \ { \left( lnx \right) }^{ 2 } } } \ dx\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> u = x</label></p><p><label class="radio"><input type="radio"> Integration by substitution will not work</label></p><p><label class="radio"><input type="radio"> u = <span class="math-tex">\(\frac { 1 }{ x } \)</span></label></p><p><label class="radio"><input class="c" type="radio"> u = lnx</label></p></div><div class="q-explanation"><p>If u = lnx</p><p>then <span class="math-tex">\(\frac { du }{ dx } =\frac { 1 }{ x }\)</span></p><p>Integrand becomes <span class="math-tex">\(\int { \frac { 3 }{ u^2 }du } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following integral can be performed by using integration by substitution (or U-substitution). Which is the correct substitution?</p><p><span class="math-tex">\(\int { 2x \ { sin3x^2 }dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> u = x<sup>2</sup></label></p><p><label class="radio"><input type="radio"> Integration by substitution will not work</label></p><p><label class="radio"><input type="radio"> u = sinx</label></p><p><label class="radio"><input class="c" type="radio"> u = 3x<sup>2</sup></label></p></div><div class="q-explanation"><p>If u = 3x<sup>2</sup></p><p>then <span class="math-tex">\(\frac { du }{ dx } =6x\)</span></p><p>Integrand becomes <span class="math-tex">\(\int { \frac { 1 }{ 3 }sinu \ du } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following integral can be performed by using integration by substitution (or U-substitution). Which is the correct substitution?</p><p><span class="math-tex">\(\int { sinx{ e }^{ cosx }dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> u = sinx</label></p><p><label class="radio"><input class="c" type="radio"> u = cosx</label></p><p><label class="radio"><input type="radio"> u = <span class="math-tex">\(e ^{ x }\)</span></label></p><p><label class="radio"><input type="radio"> Integration by substitution will not work</label></p></div><div class="q-explanation"><p>If u = cosx</p><p>then <span class="math-tex">\(\frac { du }{ dx } =-sinx\)</span></p><p>Integrand becomes <span class="math-tex">\(\int {- { e }^{ u }du } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following integral can be performed by using integration by substitution (or U-substitution). Which is the correct substitution?</p><p><span class="math-tex">\(\int { { x^2 \ lnx} \ dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> u = x</label></p><p><label class="radio"><input type="radio"> u = lnx</label></p><p><label class="radio"><input class="c" type="radio"> Integration by substitution does not work</label></p><p><label class="radio"><input type="radio"> u = <span class="math-tex">\(x ^{ 2 }\)</span></label></p></div><div class="q-explanation">This question requires Integration by parts</div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following integral can be performed by using integration by substitution (or U-substitution). Which is the correct substitution?</p><p><span class="math-tex">\(\int { \frac { 6x }{ 1+{ x }^{ 2 } } \ dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> u = 1+ x<sup>2</sup></label></p><p><label class="radio"><input type="radio"> Integation by substitution will not work</label></p><p><label class="radio"><input type="radio"> u = 6x</label></p><p><label class="radio"><input type="radio"> u = x<sup>2</sup></label></p></div><div class="q-explanation"><p>If u = 1+ x<sup>2</sup></p><p>then <span class="math-tex">\(\frac { du }{ dx } =2x\)</span></p><p>Integrand becomes <span class="math-tex">\(\int { \frac { 3 }{ u } du } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here's the chance to practise the techniques of integration by substitution</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#5dcc5c0f"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="5dcc5c0f"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Integration by Substitution2 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-168-634" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Which is the correct answer to this integral?</p><p><span class="math-tex">\(\int { 2x{ e }^{ x^2 }dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac {e ^{ x^2 } }{2} + C\)</span></label></p><p><label class="radio"><input type="radio"> None of these answers</label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(e ^{ x^2 } + C\)</span></label><label class="radio"></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2e ^{ 2x } + C\)</span></label></p></div><div class="q-explanation"><p>u = x<sup>2</sup></p><p><span class="math-tex">\(\frac { du }{ dx } =2x\)</span></p><p>du = 2x dx</p><hr class="hidden-separator"><p><span class="math-tex">\(\int { 2x{ e }^{ x^2 }dx } \)</span></p><p><span class="math-tex">\(= \int { { e }^{ u }du }\\ = { e }^{ u }+C \\={ e }^{ x^2 }+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which is the correct answer to this integral?</p><p><span class="math-tex">\(\int { sin^2x \ cosx \ dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\frac { sin^ 3x }{ 3} + 2sinxcos^2x+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { sin^ 3x }{ 3} + C\)</span></label></p><p><label class="radio"><input type="radio"> None of these answers</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\frac { cos^ 3x }{ 3} +C\)</span></label><label class="radio"></label></p></div><div class="q-explanation"><p>u = sinx</p><p><span class="math-tex">\(\frac { du }{ dx } =cosx\)</span></p><p>du = cosx dx</p><hr class="hidden-separator"><p><span class="math-tex">\(\int { sin^2x \ cosx \ dx } \)</span></p><p> <span class="math-tex">\(=\int { { u }^{ 2 }du } \\ =\frac{u^3}{3}+C\\ =\frac { sin^ 3x }{ 3} + C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which is the correct answer to this integral?</p><p><span class="math-tex">\(\int { \frac { 3 }{ x \ { \left( lnx \right) }^{ 2 } } } \ dx\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 3}{lnx}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { -3}{(lnx)^3}+C\)</span></label></p><p><label class="radio"><input type="radio"> None of these answers</label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { -3}{lnx}+C\)</span></label></p></div><div class="q-explanation"><p>u = lnx</p><p><span class="math-tex">\(\frac { du }{ dx } =\frac { 1 }{ x }\)</span></p><p><span class="math-tex">\( du= \frac { 1 }{ x } \ dx\)</span></p><p><span class="math-tex">\(\int { \frac { 3 }{ x \ { \left( lnx \right) }^{ 2 } } } \ dx\)</span></p><p><span class="math-tex">\(=\int { \frac { 3 }{ u^2 }du } \\ =\int { 3 }{ u^{-2 }du } \\ = \frac { 3u^{-1} }{-1}+C\\ =\frac { -3}{u}+C\\ =\frac { -3}{lnx}+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which is the correct answer to this integral?</p><p><span class="math-tex">\(\int { sinx{ e }^{ cosx }dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> None of these answers</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(e^{sinx}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(cosx \ e^{sinx}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-e^{cosx}+C\)</span></label></p></div><div class="q-explanation"><p>u = cosx</p><p><span class="math-tex">\(\frac { du }{ dx } =-sinx\)</span></p><p>-1du = sinx dx</p><hr class="hidden-separator"><span class="math-tex">\(\int { sinx{ e }^{ cosx }dx } \)</span><p><span class="math-tex">\(=\int {- { e }^{ u }du } \\ =-e^{cosx}+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which is the correct answer to this integral?</p><p><span class="math-tex">\(\int { \frac { 6x }{ 1+{ x }^{ 2 } } \ dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> None of these answers</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\( { -\frac { 3 }{ (1+{ x }^{ 2 })^2 } } +C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(3 \ ln|1+x^2|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\( \ ln|1+x^2|+C\)</span></label></p></div><div class="q-explanation"><p>u = 1+ x<sup>2</sup></p><p><span class="math-tex">\(\frac { du }{ dx } =2x\)</span></p><p>du = 2x dx</p><hr class="hidden-separator"><p><span class="math-tex">\(\int { \frac { 6x }{ 1+{ x }^{ 2 } } \ dx } \)</span></p><p><span class="math-tex">\(=\int { \frac { 3 }{ u } du } \\ =3 \ ln|u|+C\\ =3 \ ln|1+x^2|+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="294"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Find <span class="math-tex">\(\int { \frac { { e }^{ x } }{ 1+{ e }^{ x } } dx } \)</span></p> <p>b) Evaluate <span class="math-tex">\(\int _{ \frac { { \pi }^{ 2 } }{ 4 } }^{ \pi ^{ 2 } }{ \frac { cos\sqrt { x } }{ \sqrt { x } } } dx\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) This question requires us to use integration by substitution (or recognition)</p> <p>u =1 + e<sup>x</sup></p> <hr class="hidden-separator"> <p>b) This question requires us to use integration by substitution</p> <p>u = <span class="math-tex">\(\sqrt{x}\)</span></p> <content> </content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/integration/by-substitution/esq-sol-intbysub1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-substitution/esq-sol-intbysub1.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/231299430"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="296"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> The following diagram shows the graph of f(x) = <span class="math-tex">\(\frac { 4x }{ \sqrt { { x }^{ 2 }+1 } }\)</span></p> <p>Let R be the region bounded by <em>f</em>, the x-axis, x = 1 and x = 2</p> <p><img alt="" src="../../files/integration/by-substitution/esq2v2.png" style="width: 300px; height: 259px;"></p> <p>Find <em><strong>R</strong></em></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>This question requires you to how to find the area under the graph: Area = <span class="math-tex">\(\int _{ a }^{ b }{ y }\ dx\)</span></p> <p>The integral requires integration by substitution</p> <p>u = x<sup>2 </sup>+1</p> <content> </content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/integration/by-substitution/esq-sol-intbysub2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-substitution/esq-sol-intbysub2.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/232208684"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="304"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Using the fact that <span class="math-tex">\(tanx = \frac{sinx}{cosx} \)</span>, show that <span class="math-tex">\(\frac{d}{dx}(tanx) = \frac{1}{cos^2x}\)</span></p> <p>b) <strong>Hence</strong>, find <span class="math-tex">\(\int { \frac { \sqrt { tanx } }{ cos^{ 2 }x } } dx\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) This requires you to use the Quotient Rule</p> <p>b) <strong>Hence </strong>means that you should use the result from part a)</p> <p>Use Integration by substitution u = tanx</p> <content> </content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/integration/by-substitution/esq-sol-intbysub3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-substitution/esq-sol-intbysub3.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/232255854"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="298"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find <span class="math-tex">\(\int { \frac { arcsinx+x }{ \sqrt { 1-x^{ 2 } } } } dx\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>To complete this question, we need to split the integral into two parts</p> <p><span class="math-tex">\(\int { \frac { arcsinx+x }{ \sqrt { 1-x^{ 2 } } } } dx=\int { \frac { arcsinx }{ \sqrt { 1-x^{ 2 } } } } dx+\int { \frac { x }{ \sqrt { 1-x^{ 2 } } } } dx\)</span></p> <p>Both these integrals can be completed using Integration by substitution</p> <content> </content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/integration/by-substitution/esq-sol-intbysub4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-substitution/esq-sol-intbysub4.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/232260099"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="300"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Find <span class="math-tex">\(\int { sin^{ 5 }x\ dx } \)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>This might help</p> <p><span class="math-tex">\(sin^5x = sin^4x\cdot sinx = (sin^2x)^2\cdot sinx\)</span></p> <p>You will need to use Integration by substitution</p> <p>Can you think of a substitution that will work?</p> <content> </content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/integration/by-substitution/esq-sol-intbysub5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-substitution/esq-sol-intbysub5.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/232366286"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="302"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find <span class="math-tex">\(\int { \sqrt { 16-9x^{ 2 } } dx } \)</span> using the substitution 3x = 4 sin<span class="math-tex">\(\theta\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>This is a tricky question that requires very careful manipulation</p> <p>The substitution gives <span class="math-tex">\(\frac{dx}{d\theta}=\frac{4}{3}cos\theta\)</span></p> <p>The integral becomes <span class="math-tex">\(\int { \sqrt { 16-16sin^{ 2 }\theta } \cdot \frac { 4 }{ 3 } cos\theta \quad d\theta } \)</span></p> <content> </content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/integration/by-substitution/esq-sol-intbysub6.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-substitution/esq-sol-intbysub6.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/232389159"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div 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