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Equations - Homogeneous</a></label></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> Integration by Parts <a href="#" class="mark-page-favorite pull-right" data-pid="635" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../550/calculus.html">Calculus</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Integration by Parts</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/integration/by-parts/main-image.jpg" style="float: left; width: 100px; height: 100px;"></p> <p>Integration by parts is a method of integration that we use to integrate the product (usually !) of two functions. The aim is to change this product into another one that is easier to integrate. Although the formula looks quite odd at first glance, the technique is not too complicated.</p> <p style="text-align: center;"><span class="math-tex">\(\int { u \cdot\frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)</span></p> <hr class="hidden-separator"> <p>The tricky part is</p> <p>1) deciding when we use it (the question will never say ‘solve this integral by the method of integration by parts’) and</p> <p>2) deciding which part of the question we call <em><strong>u</strong></em> and which part <span class="math-tex">\(\frac { dv }{ dx }\)</span>.</p> <p>If you work through the resources on this page, you should be able to master these two things.</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 parts</li> <li>repeated integration by parts</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 following videos will help you understand all the concepts from this page</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>What and Why</p> </div> </div> <div class="panel-body"> <p>The formula for integration by parts is found from the <a href="../743/product-and-quotient-rule.html" title="Product and Quotient Rule">Product Rule</a> for differentiation. Here's how</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/229675351"></iframe></div> <p>Integration by parts is used for integrating the product of two functions (there are one or two exceptions to this that we will <a href="integration-by-parts.html#eg3">see later</a>). We use this technique when <a href="../634/integration-by-substitution.html" title="Integration by Substitution">Integration by Substitution</a> (or recognition) does not work. You can see in the table below some examples of when we use each technique</p> <table align="center" border="0" cellpadding="0" cellspacing="0" style="width: 60%;"> <thead> <tr> <th scope="col" style="text-align: center;">Integration by Substitution</th> <th scope="col" style="text-align: center;">Integration by Parts</th> </tr> </thead> <tbody> <tr> <td style="text-align: center;"><span class="math-tex">\(\int { 2x\cdot { e }^{ -3{ x }^{ 2 } }dx } \)</span></td> <td style="text-align: center;"><span class="math-tex">\(\int { 2x\cdot { e }^{ -3{ x } }dx } \)</span></td> </tr> <tr> <td style="text-align: center;"><span class="math-tex">\(\int { sinx\cdot { e }^{ cosx }dx } \)</span></td> <td style="text-align: center;"><span class="math-tex">\(\int { sinx\cdot { e }^{ x }dx } \)</span></td> </tr> <tr> <td style="text-align: center;"><span class="math-tex">\(\int { x\cdot \sqrt { x+1 } dx } \)</span></td> <td style="text-align: center;"><span class="math-tex">\(\int { x\cdot \sqrt { x+1 } dx } \)</span></td> </tr> </tbody> </table> <p>Notice that it is sometimes possible to use either technique, as is the case in the last example!</p> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-has-border panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Deciding on u and dv/dx 1</p> </div> </div> <div class="panel-body"> <div> <div> <p>When using the formula for integration by parts you need to learn which part of the product you are going to call <em><strong>u</strong></em> and which part <span class="math-tex">\(\frac { dv }{ dx }\)</span>.</p> <p style="text-align: center;"><span class="math-tex">\(\int { u \cdot \frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)</span></p> <p>Generally, you should</p> <p>a) Let <strong><em>u </em></strong>be the function that gets simpler when you differentiate</p> <p>b) Let <span class="math-tex">\(\frac { dv }{ dx }\)</span>be the function that doesn't get too complicated when you integrate</p> <p>Here is a video that will help you with that. It shows what happens when you make the correct choice and what happens when you make the wrong choice.</p> <p><strong><em>Use integration by parts to find <span class="math-tex">\(\int { 2x\cdot { e }^{ -3{ x }}dx } \)</span></em></strong></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/229817902"></iframe></div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-has-border panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Deciding on u and dv/dx 2</p> </div> </div> <div class="panel-body"> <div> <p>There following list might help you with your choice. Look at the two functions in your question. The one that appears first on the following list is the one you should choose to be <strong><em>u</em></strong></p> <ol> <li>Logs</li> <li>Inverse Trig</li> <li>Algebra</li> <li>Trig / Exponential</li> </ol> <p>Here's another example</p> <p><strong><em>Use integration by parts to find <span class="math-tex">\(\int { x^ 2\cdot lnxdx } \)</span></em></strong></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/229825285"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-has-border panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Repeating the Process 1</p> </div> </div> <div class="panel-body"> <div> <div> <p>Sometimes you have to apply the method of integration by parts more than once!</p> <p><strong><em>Use integration by parts to find <span class="math-tex">\(\int { x^{ 2 }\cdot sin\left( \frac { x }{ 2 } \right) dx } \)</span></em></strong></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/229857116"></iframe></div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-has-border panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Repeating the Process 2</p> </div> </div> <div class="panel-body"> <div> <div> <p>Sometimes integration by parts really doesn't seem to help and we end up going round in circles! Here's an example where that is the case and a clever solution to it!</p> <p><strong><em>Use integration by parts to find <span class="math-tex">\(\int { e^{ x }\cdot cos2xdx } \)</span></em></strong></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/229861102"></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>Special Integrals</p> </div> </div> <div class="panel-body"> <div> <p>Integration by parts is mainly used for integrating the product of two functions. However, we can use it to find the integral of ln(x) and arctrigonometric functions too. The second question requires integration by substitution as well as integration by parts!</p> <p><strong><em>Use integration by parts to find <span class="math-tex">\(\int { lnx } dx\)</span> and <span class="math-tex">\(\int { arcsinx } dx\)</span></em></strong></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/229824657"></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-parts/integration-by-parts--revision-notes.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/integration/by-parts/integration-by-parts--revision-notes.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"> <div> <p>This is the formula for integration by parts.</p> <p style="text-align: center;"><span class="math-tex">\(\int { u \cdot \frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)</span></p> <p>Here is a quiz that gets you to practice making the correct choice for <strong><em>u </em></strong> in the formula</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#b98e7972"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="b98e7972"> <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 Parts <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="11-165-635" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> You can evaluate <strong><em><span class="math-tex">\(\int { { x\cdot e }^{ x } } dx\)</span> </em></strong>using integration by parts</p><p style="text-align: center;"><span class="math-tex">\(\int { u \cdot \frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)</span></p><p>Which part do you use for <strong><em>u</em></strong> ?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> x</label></p><p><label class="radio"><input type="radio"> Either</label></p><p><label class="radio"><input type="radio"> Integration by parts is wrong method</label></p><p><label class="radio"><input type="radio"> e<sup>x</sup></label></p></div><div class="q-explanation"><p><strong><em>u </em></strong>should be the first type of function that you see in the list below</p><p>1) Logarithmic</p><p>2) Inverse Trigonometric</p><p>3) Algebraic</p><p>4) Trigonometric/Exponential</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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> You can evaluate <strong><em><span class="math-tex">\(\int { x^3 \cdot ln(x) } dx\)</span> </em></strong>using integration by parts</p><p style="text-align: center;"><span class="math-tex">\(\int { u \cdot \frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)</span></p><p>Which part do you use for <strong><em>u</em></strong> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> Either</label></p><p><label class="radio"></label></p><p><label class="radio"><input class="c" type="radio"> ln(x)</label></p><p><label class="radio"><input type="radio"> Integration by parts is the wrong method</label></p><p><label class="radio"><input type="radio"> x<sup>3</sup></label></p></div><div class="q-explanation"><p><strong><em>u </em></strong>should be the first type of function that you see in the list below</p><p>1) Logarithmic</p><p>2) Inverse Trigonometric</p><p>3) Algebraic</p><p>4) Trigonometric/Exponential</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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> You can evaluate <strong><em><span class="math-tex">\(\int { 1\cdot arctan(x) } dx\)</span></em></strong> using integration by parts</p><p style="text-align: center;"><span class="math-tex">\(\int { u \cdot \frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)</span></p><p>Which part do you use for <strong><em>u</em></strong> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 1</label></p><p><label class="radio"><input class="c" type="radio"> arctan(x)</label></p><p><label class="radio"><input type="radio"> either</label></p><p><label class="radio"><input type="radio"> Integration by parts is the wrong method</label></p></div><div class="q-explanation">This question is an interesting one since it doesn't look like a product of two parts. Here are <a href="integration-by-parts.html#eg3" target="_blank">a couple of solutions</a> to a question like this one</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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> You can evaluate <span class="math-tex">\(\int { 2x\cdot arctan(x) } dx\)</span><strong><em> </em></strong>using integration by parts</p><p style="text-align: center;"><span class="math-tex">\(\int { u \cdot \frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)</span></p><p>Which part do you use for <strong><em>u</em></strong> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> Integration by parts is the wrong method</label></p><p><label class="radio"><input type="radio"> Either</label></p><p><label class="radio"><input type="radio"> 2x</label></p><p><label class="radio"><input class="c" type="radio"> arctan(x)</label></p></div><div class="q-explanation"><p><strong><em>u </em></strong>should be the first type of function that you see in the list below</p><p>1) Logarithmic</p><p>2) Inverse Trigonometric</p><p>3) Algebraic</p><p>4) Trigonometric/Exponential</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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> You can evaluate <strong><em><span class="math-tex">\(\int { { sinx\cdot e }^{ x } } dx\)</span> </em></strong>using integration by parts</p><p style="text-align: center;"><span class="math-tex">\(\int { u \cdot \frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)</span></p><p>Which part do you use for <strong><em>u</em></strong> ?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> either</label></p><p><label class="radio"><input type="radio"> integration by parts is the wrong method</label></p><p><label class="radio"><input type="radio"> sinx</label></p><p><label class="radio"><input type="radio"> e<sup>x</sup></label></p></div><div class="q-explanation">In this question it doesn't matter, but you have to do integration by parts twice. The question seems to go around in circles. Here's <a href="integration-by-parts.html#eg2" target="_blank">a solution</a> to a question like this</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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> You can evaluate <strong><em><span class="math-tex">\(\int { \sin(x)\cdot \cos ^{ 2 }{ (x) } } dx\)</span> </em></strong>using integration by parts</p><p style="text-align: center;"><span class="math-tex">\(\int { u \cdot \frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)</span></p><p>Which part do you use for <strong><em>u</em></strong> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> cos<sup>2</sup>(x)</label></p><p><label class="radio"><input type="radio"> sin(x)</label></p><p><label class="radio"><input class="c" type="radio"> Integration by parts is the wrong method</label></p><p><label class="radio"><input type="radio"> either</label></p></div><div class="q-explanation">You need to use <a href="../634/integration-by-substitution.html" target="_blank">integration by substitution</a> to do this question</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 is a quiz that gets you to practise the method of integration by parts</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#a93d6d63"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="a93d6d63"> <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 parts <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="11-166-635" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find the values of <em><strong>a</strong></em> , <em><strong>b</strong></em> and <em><strong>d</strong></em> in the following solution (C is the constant of integration)</p><p><span class="math-tex">\(\int { { x }^{ 3 }\cdot ln(x) } dx\)</span> </p><p>= <span class="math-tex">\(\frac { { x }^{ 4 } }{ a } ln(x)-\int { \frac { { x }^{ b} }{ 4 } } dx\)</span></p><p>= <span class="math-tex">\(\frac { { x }^{ 4 } }{ a } ln(x)-\frac { { x }^{ 4 } }{ d } + C\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>d = <input type="text" style="height: auto;" data-c="16"> <span class="review"></span></p></div><div class="q-explanation"><p>You need to use integration by parts:</p><p>u = ln(x) <span class="math-tex">\(\frac {dv}{dx}=x^3\)</span></p><p><span class="math-tex">\(\frac {dv}{dx}=\frac {1}{x}\)</span> v = <span class="math-tex">\(\frac {x^4}{4}\)</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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Drop the correct answers inside the boxes to complete this integral</p><p> <span class="q-text-draggable draggable" draggable="true">+ C</span> <span class="q-text-draggable draggable" draggable="true">x<sup>2</sup></span> <span class="q-text-draggable draggable" draggable="true">lnx</span> <span class="q-text-draggable draggable" draggable="true">e<sup>x</sup></span> <span class="q-text-draggable draggable" draggable="true">x</span></p></div><div class="q-answer"><p><span class="math-tex">\(\int {x\cdot { e }^{ x }} dx\)</span> = <input type="text" style="height: auto;" data-c="x"> <span class="review"></span> e<sup>x</sup> - e<sup>x</sup> <input type="text" style="height: auto;" data-c="+ C"> <span class="review"></span> </p></div><div class="q-explanation"><p>You need to use integration by parts:</p><p>u = x <span class="math-tex">\(\frac {dv}{dx}=e^x\)</span></p><p><span class="math-tex">\(\frac {dv}{dx}=1\)</span> v = <span class="math-tex">\(e^x\)</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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find the values of <em><strong>a</strong></em> , <em><strong>b</strong></em> , <em><strong>d </strong></em>and <strong><em>e</em></strong> in the following solution (C is the constant of integration)</p><p><span class="math-tex">\(\int { { x }^{ 2 }\cdot sin(2x) } dx\)</span></p><p>= <span class="math-tex">\(-\frac { { x }^{ 2 } }{ 2 } cos(2x)\quad +\quad \int { { x } \cdot \ a } \ dx\)</span></p><p>= <span class="math-tex">\(-\frac { { x }^{ 2 } }{ 2 } cos(2x)\quad +\quad \frac { { x }^{ 2 } }{ b } \cdot d\quad -\quad \int { \frac { sin(2x) }{ 2 } } dx\)</span></p><p>= <span class="math-tex">\(-\frac { { x }^{ 2 } }{ 2 } cos(2x)\quad +\quad \frac { { x }^{ 2 } }{ b } \cdot d\quad +\quad \frac { cos(2x) }{ e } \quad +\quad C\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="cos(2x)"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>d = <input type="text" style="height: auto;" data-c="sin(2x)"> <span class="review"></span></p><p> e = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>In this question you have to do integration by parts 2 times!</p><p><a href="integration-by-parts.html#eg2" target="_blank">Here is a solution</a> to a very similar question.</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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Evaluate <span class="math-tex">\(\int _{ 1 }^{ e }{ ln(x) dx } \)</span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>This is a classic integral that you should try to remember. It uses integration by parts</p><p><span class="math-tex">\(\int { u \cdot\frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = lnx</td><td><span class="math-tex">\(\frac{dv}{dx}=1\)</span></td></tr><tr><td><span class="math-tex">\(\frac{du}{dx}=\frac{1}{x}\)</span></td><td>v = x</td></tr><tr><td> </td><td> </td></tr></tbody></table><p><span class="math-tex">\(\int _{ 1 }^{ e }{ ln(x)dx } ={ \left[ x\cdot ln(x)\ -\ x \right] }_{ 1 }^{ e }\)</span></p><p>= <span class="math-tex">\(\left[ e\cdot ln(e)\ -\ e \right] \ -\ \left[ 1\cdot ln(1)\ -\ 1 \right] \)</span></p><p>= <span class="math-tex">\(\left[ e\cdot 1\ -\ e \right] \ -\ \left[ 1\cdot 0\ -\ 1 \right] \)</span></p><p>= 1</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> <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> <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="286"> <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"> <span class="math-tex">\(\int { { e }^{ \frac { x }{ 2 } }sin(x)\quad dx } \)</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"><content> </content> <p>You need to use the method of integration by parts</p> <p>You should apply the method twice and the integral to find should be a multiple of the question</p> </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-parts/esq-sol-integration_by_parts1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-parts/esq-sol-integration_by_parts1.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/230121633"></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="290"> <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"> <span class="math-tex">\(\int { arctanx \ dx } \)</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>The solution requires you to use Integration by Parts</p> <p>Think of the question as <span class="math-tex">\(\int { 1\cdot arctanxdx } \)</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-parts/esq-sol-integration_by_parts2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-parts/esq-sol-integration_by_parts2.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/230247202"></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 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="291"> <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"> <span class="math-tex">\(\int { 2x\cdot arctanx \ dx } \)</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 question requires you to use Integration by Parts</p> <p>In the second part you will need to integrate a rational function. Since the degree of the numerator is equal to the degree of the denominator, you need to divide the polynomials</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-parts/esq-sol-integration_by_parts3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-parts/esq-sol-integration_by_parts3.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 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