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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"> Definite Integration <a href="#" class="mark-page-favorite pull-right" data-pid="586" 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">Definite Integration</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/definite-integral-and-area/main-1.jpg" style="float: left; width: 100px; height: 100px;">Integration is sometimes called <em>antidifferentiation, </em>as it is the opposite process of differentiation. When we find an <em><strong>indefinite integral,</strong></em> we find a function with an arbitrary constant, <strong><em>C</em></strong>. Whereas, when we find a<em><strong> definite integral</strong></em>, we find a numerical value. Definite integration has many applications. For the examination, we use it to find displacement in questions about kinematics, probabilities with continuous random variables and it is linked to work on finding areas and volumes with integration.</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>definite integrals <ul> <li><span class="math-tex">\(\int_{a}^{b} f'(x)\: dx=\: f(b)-f(a)\)</span></li> </ul> </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>In the following video, we are going to look at what definite integration is (comparing to indefinite integration) and what we use it for. We'll look at some of the many applications of definite integration.</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/220365768"></iframe></div> </div> </div> <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>How to do it</p> </div> </div> <div class="panel-body"> <div> <p>The process of finding definite integrals is relatively straightforward, but it is important to carefully use the correct notation. In the following video, we will look at two examples: <span class="math-tex">\(\int_{\frac{\pi}{2}}^{\pi} cosx\: dx\)</span> and <span class="math-tex">\(\int_{1}^{e} \frac{1}{x}\: dx\)</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/220702594"></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/definite-integral-and-area/revision-notes_definite_ingration1.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/algebra/sequences-and-series/revision-notes_arithmetic_sequences.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>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#d36fe513"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="d36fe513"> <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%;"> Definite Integration <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="11-228-586" 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"> <span class="math-tex">\(\int _{ 1 }^{ 2 }{ x^3dx=\frac { a }{ 4 } } \)</span></p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="15"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\int _{ 1 }^{ 2 }{ x^{ 3 }dx }\\= { \left[ \frac { { x }^{ 4 } }{ 4 } \right] }_{ 1 }^{ 2 }\\=\left[ \frac { { 2 }^{ 4 } }{ 4 } \right] -\left[ \frac { { 1 }^{ 4 } }{ 4 } \right] \\=\frac { 16 }{ 4 } -\frac { 1 }{ 4 } \\=\frac { 15 }{ 4 } \)</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"> Evaluate <span class="math-tex">\(\int _{ 0 }^{ 4 }{ \frac { 1 }{ \sqrt { 2x+1 } } dx } \)</span></p></div><div class="q-answer"><p>Answer = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(=\int _{ 0 }^{ 4 }{ { (2x+1) }^{ -0.5 }dx } \\ ={ \left[ \frac { { (2x+1) }^{ 0.5 } }{ 2\cdot 0.5 } \right] }_{ 0 }^{ 4 }\\ ={ \left[ \sqrt { 2x+1 } \right] }_{ 0 }^{ 4 }\\ =\sqrt { 9 } -\sqrt { 1 } \\ =3-1\\ =2\)</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"> Evaluate <span class="math-tex">\(\int _{ \pi }^{ 2\pi }{ \sin { \frac { x }{ 2 } dx } } \)</span></p></div><div class="q-answer"><p>Answer = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(={ \left[ -2cos\frac { x }{ 2 } \right] }_{ \pi }^{ 2\pi }\\ =\left[ -2cos\pi \right] -\left[ -2cos\frac { \pi }{ 2 } \right] \\ =2-0\)</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"> <span class="math-tex">\(\int _{ 4 }^{ 5 }{ \left( \frac { 1 }{ x+3 } -\frac { 1 }{ x-3 } \right) } dx=ln\left( \frac { a }{ b } \right) \)</span></p><p>Work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(={ \left[ ln(x+3)-ln(x-3) \right] }_{ 4 }^{ 5 }\\ ={ \left[ ln\left( \frac { x+3 }{ x-3 } \right) \right] }_{ 4 }^{ 5 }\\ =\left[ ln\left( \frac { 8 }{ 2 } \right) -ln\left( \frac { 7 }{ 1 } \right) \right] \\ =ln4-ln7\\ =ln\frac { 4 }{ 7 } \)</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 graph below shows the function f</p><p>Evaluate the following definite integrals</p><p><img alt="" src="../../files/integration/definite-integral-and-area/q5.png" style="width: 250px; height: 233px;"></p></div><div class="q-answer"><ol><li><span class="math-tex">\(\int _{ 1 }^{ 3 }{ f(x)dx } \)</span> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span><hr class="hidden-separator"></li><li><span class="math-tex">\(\int _{ -1 }^{ 0 }{ f(x)dx } \)</span> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span><hr class="hidden-separator"></li><li> <span class="math-tex">\(\int _{ -1 }^{ 4 }{ f(x)dx } \)</span> = <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span></li></ol></div><div class="q-explanation"><ol><li><img alt="" src="../../files/integration/definite-integral-and-area/q5a.png" style="width: 150px; height: 122px;"></li><li><img alt="" src="../../files/integration/definite-integral-and-area/q5b.png" style="width: 150px; height: 116px;"> definite integral is negative as it is below x axis</li><li><img alt="" src="../../files/integration/definite-integral-and-area/q5c.png" style="width: 150px; height: 120px;"> answer = -1 + 3 + (-0.5)</li></ol></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 graph below shows the function f</p><p><span class="math-tex">\(\int _{ a }^{ 7 }{ f(x)dx } =0\)</span></p><p>Find 2 possible values of a , a<sub>1</sub> and a<sub>2</sub></p><p><img alt="" src="../../files/integration/definite-integral-and-area/q6_1.jpg" style="width: 300px; height: 154px;"></p></div><div class="q-answer"><p><span class="math-tex">\(a_1\)</span> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p><span class="math-tex">\(a_2\)</span> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>You must choose a value for <strong><em>a</em></strong> that makes the definite integral from <strong><em>a</em></strong> to 7 equal 0.</p><p>Choose a value of <strong><em>a</em></strong>, so that the area of the region <strong>below the x axis</strong> must equal the area <strong>above the x axis</strong>.</p><ol></ol></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>Given that <span class="math-tex">\(\int _{ 1 }^{ 3 }{ f(x) } dx=6\)</span> and <span class="math-tex">\(\int _{ 1 }^{ 3 }{ g(x) } dx=4\)</span></p><p>find the value of the following definite integrals</p></div><div class="q-answer"><ol><li><span class="math-tex">\(\int _{ 1 }^{ 3 }{ (f(x) } +g(x))dx\)</span> = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span><hr class="hidden-separator"></li><li> <span class="math-tex">\(\int _{ 3 }^{ 3 }{ g(x) } dx\)</span> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span><hr class="hidden-separator"></li><li><span class="math-tex">\(\int _{ 3 }^{ 1 }{ f(x) } dx\)</span> = <input type="text" style="height: auto;" data-c="-6"> <span class="review"></span></li></ol></div><div class="q-explanation"><ol><li><span class="math-tex">\(\int _{ a }^{ b }{ (f(x) } +g(x))dx=\)</span><span class="math-tex">\(\int _{ a }^{ b }{ f(x) } dx+\int _{ a }^{ b }{ g(x) } dx\)</span></li><li><span class="math-tex">\(\int _{ a }^{ a}{ g(x) } dx=0\)</span></li><li><span class="math-tex">\(\int _{ a }^{ b }{ f(x) } dx=-\int _{ b }^{ a }{ f(x) } dx\)</span></li></ol></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>Given that <span class="math-tex">\(\int _{ 2}^{ 4 }{ f(x) } dx=10\)</span> and <span class="math-tex">\(\int _{ 0}^{ 2 }{ f(x) } dx=4\)</span></p><p>find the value of the following definite integrals</p></div><div class="q-answer"><ol><li><span class="math-tex">\(\int _{ 0}^{ 4 }{ f(x) } dx\)</span> = <input type="text" style="height: auto;" data-c="14"> <span class="review"></span><hr class="hidden-separator"></li><li><span class="math-tex">\(\int _{ 2}^{ 0 }{ f(x) } dx\)</span> = <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span><hr class="hidden-separator"></li><li> <span class="math-tex">\(\int _{ 2}^{ 4 }{2 f(x) } dx\)</span> = <input type="text" style="height: auto;" data-c="20"> <span class="review"></span></li></ol></div><div class="q-explanation"><ol><li><span class="math-tex">\(\int _{ a }^{ c }{ f(x) } dx=\int _{ a }^{ b }{ f(x) } dx+\int _{ b }^{ c }{ f(x) } dx\)</span></li><li><span class="math-tex">\(\int _{ a }^{ b }{ f(x) } dx=-\int _{ b }^{ a }{ f(x) } dx\)</span></li><li><span class="math-tex">\(\int _{ a }^{ b }{k f(x) } dx=k\int _{ a }^{ b }{ f(x) } dx\)</span></li></ol></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>Given that <span class="math-tex">\(\int _{ 1}^{ 3 }{ 4f(x) } dx=20\)</span> </p><p>find the value of the following definite integrals</p></div><div class="q-answer"><ol><li><span class="math-tex">\(\int _{ 3}^{ 1 }{ f(x) } dx\)</span> = <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span><hr class="hidden-separator"></li><li> <span class="math-tex">\(\int _{ 3}^{ 3 }{ f(x) } dx\)</span> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span><hr class="hidden-separator"></li><li> <span class="math-tex">\(\int _{ 1}^{ 3 }{ (f(x)+2x) } dx\)</span> = <input type="text" style="height: auto;" data-c="13"> <span class="review"></span></li></ol></div><div class="q-explanation"><ol><li><span class="math-tex">\(\int _{ 1}^{ 3 }{ f(x) } dx\)</span> = 5 , therefore <span class="math-tex">\(\int _{ 3}^{ 1 }{ f(x) } dx\)</span> = -5</li><li><span class="math-tex">\(\int _{ a }^{ a}{ f(x) } dx=0\)</span></li><li><span class="math-tex">\(\int _{ 1}^{ 3 }{ f(x) } dx+\int _{ 1}^{ 3 }{ 2x } dx\)</span></li></ol><p>= 5 + <span class="math-tex">\({ \left[ { x }^{ 2 } \right] }_{ 1 }^{ 3 }\)</span> = 5 + [9 - 1]</p><p>= 13</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>Given that <span class="math-tex">\(\int _{ -2 }^{ 2 }{ f(x)dx=5 } \)</span> and <span class="math-tex">\(\int _{ -2 }^{ 0 }{ f(x)dx=3 } \)</span> work out</p></div><div class="q-answer"><p>a) <span class="math-tex">\(\int _{ 0 }^{ 2 }{ 2f(x)dx } \)</span> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><hr class="hidden-separator"><p>b) <span class="math-tex">\(\int _{ 2 }^{ 4 }{ f(x-2)dx } \)</span> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><hr class="hidden-separator"><p>c) <span class="math-tex">\(\int _{ -2 }^{ 2 }{ (f(x)+2)dx } \)</span> = <input type="text" style="height: auto;" data-c="13"> <span class="review"></span></p></div><div class="q-explanation"><p>a) Consider the same definite integral split into two parts</p><p><span class="math-tex">\(\int _{ -2 }^{ 0 }{ f(x)dx}+\int _{ 0 }^{ 2 }{ f(x)dx } =\int _{ -2 }^{ 2 }{ f(x)dx } \\3 + 2 = 5\\\int _{ 0 }^{ 2 }{ f(x)dx }=2\)</span></p><p>b) f(x - 2) translates the graph of f(x) 2 units to the right. This is the same as <span class="math-tex">\(\int _{ 0 }^{ 2 }{ f(x)dx } \)</span></p><p><img alt="" src="../../files/integration/definite-integral-and-area/q10b.jpg" style="width: 400px; height: 166px;"></p><p>c) f(x) + 2 translates the graph 2 units up. It creates a region the same as <span class="math-tex">\(\int _{ -2 }^{ 2 }{ f(x)dx=5 } \)</span> , but with a rectangle below. The area of the rectangle is 8.</p><p><img alt="" src="../../files/integration/definite-integral-and-area/q10c.jpg" style="width: 400px; height: 207px;"></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> <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="538"> <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>Given that <span class="math-tex">\(\int _{ 4 }^{ 8 }{ \frac { 1 }{ 2x-4 } dx= } ln\sqrt { a } \)</span> , find the value of <strong><em>a</em></strong></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><span class="math-tex">\(\int { \frac { 1 }{ ax+b } dx=\frac { 1 }{ a } ln(ax+b)+C } \)</span></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/definite-integral-and-area/esq_calculus_definiteintegral1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/definite-integral-and-area/esq_calculus_definiteintegral1.pdf" width="640"></iframe></p> </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="539"> <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"></p> <p>Consider a function f(x) such that <span class="math-tex">\(\int _{ 0 }^{ 4 }{ f(x)dx } \)</span> = 6</p> <p>Find</p> <p>a) <span class="math-tex">\(\int _{ 0 }^{ 4 }{3 f(x)dx } \)</span></p> <p>b) <span class="math-tex">\(\int _{ 0 }^{ 4 }{[ f(x)+3]dx } \)</span></p> <p>c) <span class="math-tex">\(\int _{ -3 }^{ 1 }{\frac{1}{3} f(x+3)dx } \)</span></p> <p>d) <span class="math-tex">\(\int _{ 0 }^{ 4 }{ [f(x)+x]dx } \)</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><content> </content>The following properties will be useful</p> <p><span class="math-tex">\(\int _{ a }^{ b }{k f(x)dx } =k\int _{ a }^{ b }{ f(x)dx } \)</span></p> <p><span class="math-tex">\(\int _{ a }^{ b }{ [f(x)+g(x)]dx } =\int _{ a }^{ b }{ f(x)dx } +\int _{ a }^{ b }{ g(x)dx } \)</span></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/definite-integral-and-area/esq_calculus_definiteintegral2a.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/definite-integral-and-area/esq_calculus_definiteintegral2a.pdf" width="640"></iframe></p> </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="537"> <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"></p> <p>Given that <span class="math-tex">\(\int _{ 2 }^{ 5 }{ ln(sinx)dx=A } \)</span></p> <p>show that <span class="math-tex">\(\int _{ 2 }^{ 5 }{ ln({ e }^{ x }sinx)dx=A } +\frac { 21 }{ 2 } \)</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><span class="math-tex">\(\int _{ 2 }^{ 5 }{ ln({ e }^{ x }sinx)dx= } \int _{ 2 }^{ 5 }{ ln({ e }^{ x })dx+\int _{ 2 }^{ 5 }{ ln(sinx)dx } } \)</span></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/definite-integral-and-area/esq_calculus_definiteintegral3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/definite-integral-and-area/esq_calculus_definiteintegral3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="586"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label 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