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fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/integration/mixed-integration/main_mixed_integration-1.png" style="float: left; width: 100px; height: 100px;"></p> <p>This page is ideal for practising all the skills of integration. You may wish to use this page in preparation for a test on this topic or for the final examinations. The quizzes on this page have been carefully created to take you through all the skills that you need. If you want a more in depth look, then you should go to the individual pages on these topics.</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 can practise questions on </p> <ul> <li>standard integrals</li> <li>Integration by recognition</li> <li>integration by substitution</li> <li>definite integrals</li> <li>areas between graphs</li> <li>kinematics</li> </ul> </div> <div class="panel-footer"> <div> </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's a quiz that practises the <em><strong>Standard Integrals</strong></em>: <span class="math-tex">\((ax+b)^n , e^{ax}, sin(ax+b), cos(ax+b) \)</span></p> <div> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#59b9153e"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="59b9153e"> <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%;"> Mixed SL Integration 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="11-392-1165" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\int { 2sin(3x+1)dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 2cos(3x+1) }{ 3 } +C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-\frac { 2cos(3x+1) }{ 3 } +C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(6cos(3x+1)+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-6cos(3x+1)+C\)</span></label></p></div><div class="q-explanation"><p>Use the standard result <span class="math-tex">\(\int { sin(ax+b)dx } =-\frac { cos(ax+b) }{ a } +C\)</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>Work out <span class="math-tex">\(\int { \frac { cos(0.5x-2) }{ 2 } dx } \)</span></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(sin(0.5x-2)+C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-\frac { sin(0.5x-2) }{ 4 } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-sin(0.5x-2)+C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { sin(0.5x-2) }{ 4 } +C\)</span></span></label> </p></div><div class="q-explanation"><p>Use the standard result <span class="math-tex">\(\int { cos(ax+b)dx } =\frac { sin(ax+b) }{ a } +C\)</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>Work out <span class="math-tex">\(\int { { e }^{ 2x-1} }dx\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(e^{x^2-x}+C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(e^{2x-1}+C\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac { { e }^{ 2x-1 } }{ 2 } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(2{ e }^{ 2x-1}+C\)</span></span></label> </p></div><div class="q-explanation"><p>Use the standard result <span class="math-tex">\(\int { { e }^{ ax+b }dx=\frac { { e }^{ ax+b } }{ a } } +C\)</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>Work out <span class="math-tex">\(\int { \frac { 1 }{ 2x+3 }} dx\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-\frac { 2 }{ { (2x+1) }^{ 2 } } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(ln\left| 2x+3 \right| +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { 1 }{ 2 } { (2x+1) }^{ 0 }+C\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac { 1 }{ 2 } ln\left| 2x+3 \right| +C\)</span></span></label> </p></div><div class="q-explanation"><p>Use the standard result <span class="math-tex">\(\int { \frac { 1 }{ ax+b } dx=\frac { 1 }{ a } } ln\left| ax+b \right| +C\)</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>Work out <span class="math-tex">\(\int { (3x-1)^{ 4 } } dx\)</span></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac { (3x-1)^{ 5 } }{ 15 } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\((3x-1)^5+C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { (3x-1)^{ 3 } }{ 9 } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\((3x-1)^3+C\)</span></span></label> </p></div><div class="q-explanation"><p>Use the standard result <span class="math-tex">\(\int { (ax+b)^{ n }dx=\frac { (ax+b)^{ n+1 } }{ a(n+1) } +C } \)</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>Work out <span class="math-tex">\(\int { \frac { \sqrt { 3x+1 } }{ 2 } dx } \)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { { (3x+1) }^{ \frac { 3 }{ 2 } } }{ 3 } +C\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac { { (3x+1) }^{ \frac { 3 }{ 2 } } }{ 9 } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { 3 }{ \sqrt { 3x+1 } } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { \sqrt { { (3x+1) }^{ 3 } } }{ 2 } +C\)</span></span></label> </p></div><div class="q-explanation"><p>Use the standard result <span class="math-tex">\(\int { (ax+b)^{ n }dx=\frac { (ax+b)^{ n+1 } }{ a(n+1) } +C } \)</span></p><p><span class="math-tex">\(\int { \frac { \sqrt { 3x+1 } }{ 2 } dx } =\frac{1}{2}\int { { (3x+1) }^{ \frac{1}{2} } dx}\)</span></p><p><span class="math-tex">\(=\frac { 1 }{ 2 } \frac { { (3x+1) }^{ \frac { 3 }{ 2 } } }{ 3\left( \frac { 3 }{ 2 } \right) } +C\)</span></p><p><span class="math-tex">\(=\frac { { (3x+1) }^{ \frac { 3 }{ 2 } } }{ 9 } +C\)</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>Work out <span class="math-tex">\(\int { \frac { 3 }{ { (2+5x) }^{ 2 } } dx } \)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-\frac { 15 }{ (2+5x) } +C\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(-\frac { 3 }{ 5(2+5x) } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-\frac { 1 }{5 (2+5x)^3 } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { 5 }{ (2+5x)^3 } +C\)</span></span></label> </p></div><div class="q-explanation"><p>Use the standard result <span class="math-tex">\(\int { (ax+b)^{ n }dx=\frac { (ax+b)^{ n+1 } }{ a(n+1) } +C } \)</span></p><p><span class="math-tex">\(\int { \frac { 3 }{ { (2+5x) }^{ 2 } } dx } =3\int { (2+5x)^{ -2 }dx } \)</span></p><p><span class="math-tex">\(=3\frac { { (2+5x) }^{ -1 } }{ (-1)5 } +C\)</span></p><p><span class="math-tex">\(-\frac { 3 }{ 5(2+5x) } +C\)</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>Work out <span class="math-tex">\(\int { \frac { 4 }{ \sqrt { 2x-1 } } dx } \)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\sqrt { 2x-1 } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-\frac { 4 }{ 3 } { (2x-1) }^{ -\frac { 3 }{ 2 } }+C\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(4\sqrt { 2x-1 } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-\frac { 4 }{ { (2x-1) }^{ \frac { 3 }{ 2 } } } +C\)</span></span></label> </p></div><div class="q-explanation"><p>Use the standard result <span class="math-tex">\(\int { (ax+b)^{ n }dx=\frac { (ax+b)^{ n+1 } }{ a(n+1) } +C } \)</span></p><p><span class="math-tex">\(\int { \frac { 4 }{ \sqrt { 2x-1 } } dx } =4\int { (2x-1)^{ -\frac { 1 }{ 2 } }dx } \)</span></p><p><span class="math-tex">\(=4\frac { { (2x-1) }^{ \frac { 1 }{ 2 } } }{ (2)\left( \frac { 1 }{ 2 } \right) } +C\)</span></p><p><span class="math-tex">\(=4\sqrt { 2x-1 } +C\)</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>Work out <span class="math-tex">\(\int { \frac { 4 }{ { e }^{ 3x } } dx }\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { 4 }{ 3{ e }^{ 3x } } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { 4 }{ 3}{ e }^{ 3x } +C\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(-\frac { 4 }{ 3{ e }^{ 3x } } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { 12 }{ { e }^{ 3x } } +C\)</span></span></label> </p></div><div class="q-explanation"><p>Use the standard result <span class="math-tex">\(\int { { e }^{ ax+b }dx=\frac { { e }^{ ax+b } }{ a } } +C\)</span></p><p><span class="math-tex">\(\int { \frac { 4 }{ { e }^{ 3x } } dx } =4\int { { e }^{ -3x }dx}\)</span></p><p><span class="math-tex">\(=4\frac { { e }^{ -3x }}{ { -3 } } +C\)</span></p><p><span class="math-tex">\(=-\frac { 4 }{ 3{ e }^{ 3x } } +C\)</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>Work out <span class="math-tex">\(\int { \sqrt { { e }^{ 4x } } dx } \)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-\frac { 1 }{ 4\sqrt { { e }^{ 4x } } } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { 1 }{ \sqrt { { e }^{ 4x } } } +C\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-2{ e }^{ -2x }+C\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac { { e }^{ 2x } }{ 2 } +C\)</span></span></label> </p></div><div class="q-explanation"><p>Use the standard result <span class="math-tex">\(\int { { e }^{ ax+b }dx=\frac { { e }^{ ax+b } }{ a } } +C\)</span></p><p><span class="math-tex">\(\int { \sqrt { { e }^{ 4x } } dx } =\int { { \left( { e }^{ 4x } \right) }^{ \frac { 1 }{ 2 } }dx } \)</span></p><p><span class="math-tex">\(=\int { { { e }^{ 2x } }dx } \)</span></p><p><span class="math-tex">\(=\frac { { e }^{ 2x } }{ 2 } +C\)</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> <hr class="hidden-separator"> <p>Here is a quiz that practises <em><strong>Integration by Recognition in the form <span class="math-tex">\(\large\int{f'(x)e^{f(x)}{dx}}\)</span></strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#d1a000f8"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="d1a000f8"> <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 recognition 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="11-604-1165" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int{2xe^{x^2}{dx}}\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large e^{x^2}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2x^{-2}e^{x^3}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{e^{x^3}}{3x^2}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2e^{x^2}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large\int{f'(x)e^{f(x)}{dx}}=e^{f(x)}+C\)</span></p><p>We can also use integration by substitution with the substitution <span class="math-tex">\(u=x^2\)</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>Work out <span class="math-tex">\(\large\int{\cos xe^{\sin x}{dx}}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large e^{\cos x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large e^{\sin 2x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large -e^{\sin x}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large e^{\sin x}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large\int{f'(x)e^{f(x)}{dx}}=e^{f(x)}+C\)</span></p><p>We can also use integration by substitution with the substitution <span class="math-tex">\(u=\sin x\)</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>Work out <span class="math-tex">\(\large\int{\sin xe^{\cos x}{dx}}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \cos xe^{\sin x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \tan xe^{\cos x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large e^{\cos x}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large -e^{\cos x}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large\int{f'(x)e^{f(x)}{dx}}=e^{f(x)}+C\)</span></p><p>We can also use integration by substitution with the substitution <span class="math-tex">\(u=\cos x\)</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>Work out <span class="math-tex">\(\large\int{x^2e^{x^3}{dx}}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large x^2e^{x^3}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{e^{x^3}}{x}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{1}{3}e^{x^3}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{x^2e^{x^3}}{3}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large\int{f'(x)e^{f(x)}{dx}}=e^{f(x)}+C\)</span></p><p>We can also use integration by substitution with the substitution <span class="math-tex">\(u=x^3\)</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>Work out <span class="math-tex">\(\large\int{x^3e^{x^4}{dx}}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{x^3}{4}e^{x^4}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 4e^{x^4}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large e^{x^4}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{1}{4}e^{x^4}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large\int{f'(x)e^{f(x)}{dx}}=e^{f(x)}+C\)</span></p><p>We can also use integration by substitution with the substitution <span class="math-tex">\(u=x^4\)</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>Work out <span class="math-tex">\(\large\int{(x+1)e^{x^2+2x}{dx}}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2e^{x^2+2x}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{1}{2}e^{x^2+2x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large (x+1)e^{x^2+2x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large (x^2+2x)e^{x^2+2x}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large\int{f'(x)e^{f(x)}{dx}}=e^{f(x)}+C\)</span></p><p>We can also use integration by substitution with the substitution <span class="math-tex">\(u=x^2+2x\)</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>Work out <span class="math-tex">\(\large\int{\frac{e^{\sqrt{x}}}{\sqrt{x}}{dx}}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large -2e^{\sqrt{x}}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large 2e^{\sqrt{x}}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large -\frac{1}{2}e^{\sqrt{x}}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{1}{2}e^{\sqrt{x}}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large\int{f'(x)e^{f(x)}{dx}}=e^{f(x)}+C\)</span></p><p>We can also use integration by substitution with the substitution <span class="math-tex">\(u=x^{0.5}\)</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>Work out <span class="math-tex">\(\large\int{\cos 2x\ e^{\sin 2x}{dx}}\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{1}{2}e^{\sin 2x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large e^{\sin 2x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2e^{\cos 2x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{1}{2}e^{\cos 2x}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large\int{f'(x)e^{f(x)}{dx}}=e^{f(x)}+C\)</span></p><p>We can also use integration by substitution with the substitution <span class="math-tex">\(u=\sin 2x\)</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>Work out <span class="math-tex">\(\large\int{(1+\ln x)e^{x \ln x}{dx}}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large (1+\ln x)^2e^{x\ln x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large e^{1+\ln x}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large e^{x \ln x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large xe^{x \ln x}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large\int{f'(x)e^{f(x)}{dx}}=e^{f(x)}+C\)</span></p><p>We can also use integration by substitution with the substitution <span class="math-tex">\(u=x\ln x\)</span></p><p>We use the product rule to differentiate <span class="math-tex">\(x\ln x\)</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>Work out <span class="math-tex">\(\large\int{x \sin x^2\cdot e^{\cos x^2}{dx}}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large -2e^{\cos x^2}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 0.5e^{\cos x^2}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large -0.5e^{\cos x^2}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2e^{\cos x^2}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large\int{f'(x)e^{f(x)}{dx}}=e^{f(x)}+C\)</span></p><p>We can also use integration by substitution with the substitution <span class="math-tex">\(u=\cos x^2\)</span></p><p>We use the chain rule to differentiate <span class="math-tex">\(\cos x^2\)</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> </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"></div> <p>Here is a quiz that practises <em><strong>Integration by Recognition in the form</strong></em> <span class="math-tex">\(\large \int f'(x)[f(x)]^{n} {d x}\)</span></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#90aa4369"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="90aa4369"> <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 recognition 2 <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="11-605-1165" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large \int 2x(x^2+1)^3 {d x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{(x^2+1)^4}{4}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large (x^2+1)^4+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{2x(x^2+1)^4}{4}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{(x^2+1)^4}{6}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large \int f'(x)[f(x)]^{n} {d x}=\frac{[f(x)]^{n+1}}{n+1}+C\)</span></p><p>We can also use integration by substitution using the substitution <span class="math-tex">\(\large u=x^2+1\)</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>Work out <span class="math-tex">\(\large \int e^x(e^x+1)^2 {d x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large (e^x+1)^3+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2(e^x+1)^3+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{(e^x+1)^3}{3}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{e^x(e^x+1)^3}{3}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large \int f'(x)[f(x)]^{n} {d x}=\frac{[f(x)]^{n+1}}{n+1}+C\)</span></p><p>We can also use integration by substitution using the substitution <span class="math-tex">\(\large u=e^x+1\)</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>Work out <span class="math-tex">\(\large \int \cos x\sin ^2x \ {d x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \sin ^3x+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{\cos ^3x}{3}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{\sin ^3x}{3}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large -\frac{\sin ^3x}{3}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large \int f'(x)[f(x)]^{n} {d x}=\frac{[f(x)]^{n+1}}{n+1}+C\)</span></p><p>We can also use integration by substitution using the substitution <span class="math-tex">\(\large u=\sin x\)</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>Work out <span class="math-tex">\(\large \int \frac{(\ln x+1)^2}{x} {d x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{(\ln x+1)^3}{3x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large2(\ln x+1)+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large2(\ln x+1)^3+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{(\ln x+1)^3}{3}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large \int f'(x)[f(x)]^{n} {d x}=\frac{[f(x)]^{n+1}}{n+1}+C\)</span></p><p>We can also use integration by substitution using the substitution <span class="math-tex">\(\large u=\ln x+1\)</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>Work out <span class="math-tex">\(\large \int 9x^2(x^3+3)^2{d x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 9(x^3+3)^3+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large (x^3+3)^3+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 3(x^3+3)^3+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{(x^3+3)^3}{3}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large \int f'(x)[f(x)]^{n} {d x}=\frac{[f(x)]^{n+1}}{n+1}+C\)</span></p><p>We can also use integration by substitution using the substitution <span class="math-tex">\(\large u=x^3+3\)</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>Work out <span class="math-tex">\(\large \int \frac{2x}{(x^2+1)^2}{d x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{-1}{(x^2+1)^3}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{1}{x^2+1}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{-1}{x^2+1}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{1}{(x^2+1)^3}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large \int f'(x)[f(x)]^{n} {d x}=\frac{[f(x)]^{n+1}}{n+1}+C\)</span></p><p>It helps if we think about the question to be <span class="math-tex">\(\large \int \frac{2x}{(x^2+1)^2}{d x}=\int 2x(x^2+1)^{-2}{d x}\)</span></p><p>We can also use integration by substitution using the substitution <span class="math-tex">\(\large u=x^2+1\)</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>Work out <span class="math-tex">\(\large \int \frac{e^x}{(e^x-1)^3}{d x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{-4}{(e^x-1)^4}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{-3}{(e^x-1)^4}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{2}{(e^x-1)^2}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{-1}{2(e^x-1)^2}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large \int f'(x)[f(x)]^{n} {d x}=\frac{[f(x)]^{n+1}}{n+1}+C\)</span></p><p>It helps if we think about the question to be <span class="math-tex">\(\large \int \frac{e^x}{(e^x-1)^3}{d x}=\int e^x(e^x-1)^{-3}{d x}\)</span></p><p>We can also use integration by substitution using the substitution <span class="math-tex">\(\large u=e^x-1\)</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>Work out <span class="math-tex">\(\large \int 2x\sqrt{x^2+1}{d x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{3}{2}(x^2+1)^{\frac{3}{2}}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large (x^2+1)^{-\frac{1}{2}}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{2}{3}(x^2+1)^{\frac{3}{2}}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large (x^2+1)^{\frac{3}{2}}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large \int f'(x)[f(x)]^{n} {d x}=\frac{[f(x)]^{n+1}}{n+1}+C\)</span></p><p>It helps if we think about the question to be <span class="math-tex">\(\large \int 2x\sqrt{x^2+1}{d x}=\large \int 2x(x^2+1)^{\frac{1}{2}}{d x}\)</span></p><p>We can also use integration by substitution using the substitution <span class="math-tex">\(\large u=x^2+1\)</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>Work out <span class="math-tex">\(\large \int 3{\cos x\sqrt{\sin x}}{d x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{2}{3}\sqrt{\sin x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large -2(\sin x)^\frac{3}{2}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large 2\sqrt{\sin ^3x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large -\frac{2}{3}\sqrt{\sin x}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large \int f'(x)[f(x)]^{n} {d x}=\frac{[f(x)]^{n+1}}{n+1}+C\)</span></p><p>It helps if we think about the question to be <span class="math-tex">\(\large \int 2{\cos x\sqrt{\sin x}}{d x}= 3\int \cos x(\sin x)^{\frac{1}{2}}{d x}\)</span></p><p>We can also use integration by substitution using the substitution <span class="math-tex">\(\large u=\sin x\)</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>Work out <span class="math-tex">\(\large \int \frac{\cos 2x}{\sqrt{1-\sin 2x}}{d x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large -2{\sqrt{1-\sin 2x}}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2{\sqrt{1-\sin 2x}}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large {\sqrt{1-\sin 2x}}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large -{\sqrt{1-\sin 2x}}+C\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large \int f'(x)[f(x)]^{n} {d x}=\frac{[f(x)]^{n+1}}{n+1}+C\)</span></p><p>It helps if we think about the question to be <span class="math-tex">\(\large \int \frac{\cos 2x}{\sqrt{1-\sin 2x}}{d x}=\int \cos 2x(1-\sin 2x)^{-\frac{1}{2}}{d x}\)</span></p><p>We can also use integration by substitution using the substitution <span class="math-tex">\(\large u=1-\sin 2x\)</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 a quiz that practises <em><strong>Integration by Substitution</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#8f590cec"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="8f590cec"> <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%;"> Mixed SL Integration 2 <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="11-393-1165" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\int { x{ e }^{ x² }dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\( { e }^{ x² }+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { 1 }{ 2 } { e }^{ x² }+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(x{ e }^{ x² }+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 1 }{ 4 } { e }^{ x² }+C\)</span></label></p></div><div class="q-explanation"><p>We can use integration by substitution. Let u = x²</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int { x{ e }^{ x² }dx } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u=x²\\ \frac { du }{ dx } =2x\\ \frac { 1 }{ 2 } du=xdx\)</span></td></tr><tr><td><span class="math-tex">\(\int { { e }^{ u }\frac { 1 }{ 2 } du } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { 1 }{ 2 } \int { { e }^{ u }du } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { 1 }{ 2 } { e }^{ u }+C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { 1 }{ 2 } { e }^{ x² }+C\)</span></td><td> </td></tr><tr><td> </td><td> </td></tr></tbody></table></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>Work out <span class="math-tex">\(\int { { e }^{ x }{ \left( { e }^{ x }+1 \right) }^{ 3 }dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\({ \left( { e }^{ x }+1 \right) }^{ 4 }+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({ \left( { e }^{ x }+1 \right) }^{ 3 }+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { { \left( { e }^{ x }+1 \right) }^{ 4 } }{ 4 } +C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({ e }^{ x }{ \left( { e }^{ x }+1 \right) }^{ 3 }+C\)</span></label></p></div><div class="q-explanation"><p>We can use integration by substitution. Let <span class="math-tex">\(u={ e }^{ x }+1\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int { { e }^{ x }{ \left( { e }^{ x }+1 \right) }^{ 3 }dx } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u={ e }^{ x }+1\\ \frac { du }{ dx } ={ e }^{ x }\\ du={ e }^{ x }dx\)</span></td></tr><tr><td><span class="math-tex">\(\int { { u }^{ 3 }du } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { { u }^{ 4 } }{ 4 } +C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { { \left( { e }^{ x }+1 \right) }^{ 4 } }{ 4 } +C\)</span></td><td> </td></tr><tr><td> </td><td> </td></tr></tbody></table></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>Work out <span class="math-tex">\(\int { x\sqrt { x^{ 2 }+1 } dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 2{ (x²+1) }^{ \frac { 3 }{ 2 } } }{ 3 } +C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 1 }{ 2 } \sqrt { x^{ 2 }+1 } +C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { { (x²+1) }^{ \frac { 3 }{ 2 } } }{ 3 } +C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\frac { 1 }{ 4\sqrt { x²+1 } } +C\)</span></label></p></div><div class="q-explanation"><p>We can use integration by substitution. Let u = x²+1</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int { x\sqrt { x^{ 2 }+1 } dx } \)</span></td><td> </td></tr><tr><td> </td><td><p><span class="math-tex">\(u=x^2+1
\\
\frac { du }{ dx } =2x
\\
\frac{1}{2}du=xdx\)</span></p></td></tr><tr><td><span class="math-tex">\(\int { \sqrt { u }\cdot \frac{1}{2}du } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { 1 }{ 2 } \int { { u }^{ \frac { 1 }{ 2 } }du } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { 1 }{ 2 } \frac { { u }^{ \frac { 3 }{ 2 } } }{ \frac { 3 }{ 2 } } +C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { { u }^{ \frac { 3 }{ 2 } } }{ 3 } +C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { { (x²+1) }^{ \frac { 3 }{ 2 } } }{ 3 } +C\)</span></td><td> </td></tr></tbody></table></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>Work out <span class="math-tex">\(\int { { e }^{ x }sin\left( { e }^{ x } \right) dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(-sin({ e }^{ x })+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-cos({ e }^{ x })+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(cos({ e }^{ x })+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(sin({ e }^{ x })+C\)</span></label></p></div><div class="q-explanation"><p>We can use integration by substitution. Let <span class="math-tex">\(u={ e }^{ x }\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int { { e }^{ x }sin\left( { e }^{ x } \right) dx } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u={ e }^{ x }\\ \frac { du }{ dx } ={ e }^{ x }\\ du={ e }^{ x }dx\)</span></td></tr><tr><td><span class="math-tex">\(\int { sinudu } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=-cosu+C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=-cos({ e }^{ x })+C\)</span></td><td> </td></tr><tr><td> </td><td> </td></tr></tbody></table></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>Work out <span class="math-tex">\(\int { \frac { 2x+3 }{ x²+3x } dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 2x+3 }{ (x²+3x)² } +C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(ln|x²+3x|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 1 }{ x²+3x } +C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{2}ln|x²+3x|+C\)</span></label></p></div><div class="q-explanation"><p>We can use integration by substitution. Let <span class="math-tex">\(u=x²+3x\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int { \frac { 2x+3 }{ x²+3x } dx } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u=x²+3x\\ \frac { du }{ dx } =2x+3\\ du=\left( 2x+3 \right) dx\)</span></td></tr><tr><td><span class="math-tex">\(\int { \frac { 1 }{ u } du } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=ln|u|+C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=ln|x²+3x|+C\)</span></td><td> </td></tr></tbody></table></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>Work out <span class="math-tex">\(\int { cosxsin²xdx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(sin²x+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { sin^{ 3 }x }{ 3 } +C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\frac { cos^{ 3 }x }{ 3 } +C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\frac { sin^{ 3 }x }{ 3 } +C\)</span></label></p></div><div class="q-explanation"><p>We can use integration by substitution. Let <span class="math-tex">\(u=sinx\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int { cosxsin²xdx } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u=sinx\\ \frac { du }{ dx } =cosx\\ du=cosxdx\)</span></td></tr><tr><td><span class="math-tex">\(\int { u²du } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { { u }^{ 3 } }{ 3 } +C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { sin^{ 3 }x }{ 3 } +C\)</span></td><td> </td></tr></tbody></table></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>Work out <span class="math-tex">\(\int { \frac { cosx }{ sinx } dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(ln|cosx|+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(ln|sinx|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(sec²x+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-ln|sinx|+C\)</span></label></p></div><div class="q-explanation"><p>We can use integration by substitution. Let <span class="math-tex">\(u=sinx\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int { \frac { cosx }{ sinx } dx } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u=sinx\\ \frac { du }{ dx } =cosx\\ du=cosxdx\)</span></td></tr><tr><td><span class="math-tex">\(\int { \frac { 1 }{ u } du } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=ln|u|+C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=ln|sinx|+C\)</span></td><td> </td></tr></tbody></table></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>Work out <span class="math-tex">\(\int { \frac { lnx }{ x } dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { { \left( lnx \right) }^{ 2 } }{ 2 } +C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(ln|x|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { lnx }{ x^2 }+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\((ln|x|)^2+C\)</span></label></p></div><div class="q-explanation"><p>We can use integration by substitution. Let <span class="math-tex">\(u=lnx\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int { \frac { lnx }{ x } dx } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u=lnx\\ \frac { du }{ dx } =\frac { 1 }{ x } \\ du=\frac { 1 }{ x } dx\)</span></td></tr><tr><td><span class="math-tex">\(\int { udu } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { { u }^{ 2 } }{ 2 } +C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { { \left( lnx \right) }^{ 2 } }{ 2 } +C\)</span></td><td> </td></tr></tbody></table></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>Work out <span class="math-tex">\(\int { \frac { { e }^{ \sqrt { x } } }{ \sqrt { x } } dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\({ e }^{ \sqrt { x } }+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{2}{ e }^{ \sqrt { x } }+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(2{ e }^{ \sqrt { x } }+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-2{ e }^{ \sqrt { x } }+C\)</span></label></p></div><div class="q-explanation"><p>We can use integration by substitution. Let <span class="math-tex">\(u={ x }^{ \frac { 1 }{ 2 } }\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int { \frac { { e }^{ \sqrt { x } } }{ \sqrt { x } } dx } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u={ x }^{ \frac { 1 }{ 2 } }\\ \frac { du }{ dx } =\frac { 1 }{ 2 } { x }^{ -\frac { 1 }{ 2 } }\\ \frac { du }{ dx } =\frac { 1 }{ 2 } \frac { 1 }{ \sqrt { x } } \\ 2du=\frac { 1 }{ \sqrt { x } } dx\)</span></td></tr><tr><td><span class="math-tex">\(2\int { { e }^{ u }dx } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=2{ e }^{ u }+C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=2{ e }^{ \sqrt { x } }+C\)</span></td><td> </td></tr></tbody></table></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>Work out <span class="math-tex">\(\int { { cos }^{ 3 }xdx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(1-sin²x+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(sinx-\frac { sin^{ 3 }x }{ 3 } +C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{{ sin }^{ 4 }x}{4}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { sin^{ 3 }x }{ 3 } -sinx+C\)</span></label></p></div><div class="q-explanation"><p>This question becomes more obvious if we re-write the integrand</p><p><span class="math-tex">\(\int { { cos }^{ 3 }xdx } =\int { { cosxcos }^{ 2 }xdx } \)</span></p><p>Use a Pythagorean Trig Identity to write cos²x in terms of sin²x</p><p><span class="math-tex">\(=\int { { cosx(1-sin }^{ 2 }x)dx } \)</span></p><p>Now, we can use integration by substitution. Let <span class="math-tex">\(u=sinx\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int { { cosx(1-sin }^{ 2 }x)dx } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u=sinx\\ \frac { du }{ dx } =cosx\\ du=cosxdx\)</span></td></tr><tr><td><span class="math-tex">\(\int { (1-u²)du } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=u-\frac { { u }^{ 3 } }{ 3 } +C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=sinx-\frac { sin^{ 3 }x }{ 3 } +C\)</span></td><td> </td></tr></tbody></table></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 a quiz that practises <em><strong>Definite Integration</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#d03dfc3d"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="d03dfc3d"> <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%;"> Mixed SL Integration 3 <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="11-394-1165" 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"> Work out <span class="math-tex">\(\int _{ \frac { \pi }{ 4 } }^{ \frac { \pi }{ 2 } }{ sin2xdx } \)</span></p></div><div class="q-answer"><p>Answer = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\({ \left[ \frac { -cos2x }{ 2 } \right] }_{ \frac { \pi }{ 4 } }^{ \frac { \pi }{ 2 } }\)</span></p><p><span class="math-tex">\(=\frac { -cos\pi }{ 2 } -\frac { -cos\left( \frac { \pi }{ 2 } \right) }{ 2 } \)</span></p><p><span class="math-tex">\(=-\frac { -1 }{ 2 } -0\)</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><span class="math-tex">\(\int _{ -1 }^{ 1 }{ \frac { 1 }{ x+2 } dx } =lnk\)</span></p><p>Work out the value of <strong><em>k</em></strong></p></div><div class="q-answer"><p><strong><em>k</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\({ \left[ ln|x+2| \right] }_{ -1 }^{ 1 }\)</span></p><p><span class="math-tex">\(=ln|3|-ln|1|\)</span></p><p><span class="math-tex">\(=ln3\)</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"> Work out <span class="math-tex">\(\int _{ 1 }^{ 4 }{ \frac { 1 }{ \sqrt { x } } dx } \)</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">\(\int _{ 1 }^{ 4 }{ \frac { 1 }{ \sqrt { x } } dx } =\int _{ 1 }^{ 4 }{ { x }^{ -0.5 }dx } \)</span></p><p><span class="math-tex">\( ={ \left[ \frac { { x }^{ 0.5 } }{ 0.5 } \right] }_{ 1 }^{ 4 }\)</span></p><p><span class="math-tex">\(={ \left[ 2\sqrt { x } \right] }_{ 1 }^{ 4 }\)</span></p><p><span class="math-tex">\(=2\sqrt { 4 } -2\sqrt { 1 } \)</span></p><p>= 4 - 2 = 2</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><span class="math-tex">\(\int _{ -2 }^{ 0 }{ \frac { 1 }{ { e }^{ 2x } } dx } =\frac { { e }^{ a }-1 }{ 2 } \)</span></p><p>Work out the value of <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="4"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\int _{ -2 }^{ 0 }{ \frac { 1 }{ { e }^{ 2x } } dx } =\int _{ -2 }^{ 0 }{ { e }^{ -2x }dx } \)</span></p><p><span class="math-tex">\(={ \left[ \frac { { e }^{ -2x } }{ -2 } \right] }_{ -2 }^{ 0 }\)</span></p><p><span class="math-tex">\(=\frac { { -e }^{ 0 } }{ 2 } -\frac { { -e }^{ 4 } }{ 2 }\)</span></p><p><span class="math-tex">\( =-\frac { 1 }{ 2 } +\frac { { e }^{ 4 } }{ 2 } \)</span></p><p><span class="math-tex">\(=\frac { { e }^{ 4 }-1 }{ 2 } \)</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>Work out <span class="math-tex">\(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ cosx{ e }^{ sinx }dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\( { e ^{\frac{\pi}{6}}} -1\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\sqrt { e } -1\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\sqrt { e } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\( { e ^{\frac{\sqrt{3}}{2}}} -1\)</span></label></p></div><div class="q-explanation"><p>This question requires integration by substitution. Let = sinx</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ cosx{ e }^{ sinx }dx } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u=sinx\\ \frac { du }{ dx } =cosx\\ du=cosxdx\)</span></td></tr><tr><td><span class="math-tex">\(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ cosx{ e }^{ sinx }dx } \)</span></td><td> </td></tr><tr><td> </td><td>Change the limits using the substitution</td></tr><tr><td><span class="math-tex">\(=\int _{ sin0 }^{ sin\frac { \pi }{ 6 } }{ { e }^{ u }du } \)</span></td><td> </td></tr><tr><td> </td><td> </td></tr><tr><td><span class="math-tex">\(={ [{ e }^{ u }] }_{ 0 }^{ 0.5 }\)</span></td><td> </td></tr><tr><td> </td><td> </td></tr><tr><td><span class="math-tex">\(={ e }^{ 0.5 }-{ e }^{ 0 }\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\sqrt { e } -1\)</span></td><td> </td></tr></tbody></table></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><span class="math-tex">\(\int _{ e }^{ e² }{ \frac { 1 }{ xlnx } dx } =lna\)</span></p><p>Work out the value of <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="2"> <span class="review"></span></p></div><div class="q-explanation"><p>This question requires integration by substitution. Let = lnx</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int _{ e }^{ e² }{ \frac { 1 }{ xlnx } dx } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u=lnx\\ \frac { du }{ dx } =\frac { 1 }{ x } \\ du=\frac { 1 }{ x } dx\)</span></td></tr><tr><td><span class="math-tex">\(=\int _{ lne }^{ lne² }{ \frac { 1 }{ u } du } \)</span></td><td> </td></tr><tr><td> </td><td> </td></tr><tr><td><span class="math-tex">\(=\int _{ 1 }^{ 2 }{ \frac { 1 }{ u } du } \)</span></td><td> </td></tr><tr><td> </td><td> </td></tr><tr><td><span class="math-tex">\(=[{ lnu] }_{ 1 }^{ 2 }\)</span></td><td> </td></tr><tr><td> </td><td> </td></tr><tr><td><span class="math-tex">\(=ln2-ln1\)</span></td><td>ln1 = 0</td></tr><tr><td><span class="math-tex">\(=ln2\)</span></td><td> </td></tr></tbody></table></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 of the following are true?</p><p>Select <strong>ALL </strong>correct answers</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\int _{ a }^{ a }{ f(x)dx } =0\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\int _{ b }^{ a }{ f(x)dx } =-\int _{ a }^{ b }{ f(x)dx } \)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\int _{ 1 }^{ 3 }{ f(x)dx } =\int _{ 2 }^{ 3 }{ f(x)dx } +\int _{ 2 }^{ 1 }{ f(x)dx } \)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(2\int _{ 1 }^{ 2 }{ f(x)dx } =\int _{ 2 }^{ 4 }{ f(x)dx } \)</span></label></p></div><div class="q-explanation"><p>There are two incorrect answers. We could be confusing them with the following correct properties of the definite integral</p><ul><li><span class="math-tex">\(2\int _{ 1 }^{ 2 }{ f(x)dx } =\int _{ 1 }^{ 2 }{2 f(x)dx } \)</span></li><li><span class="math-tex">\(\int _{ 1 }^{ 3 }{ f(x)dx } =\int _{ 2 }^{ 3 }{ f(x)dx } +\int _{ 1 }^{2 }{ f(x)dx } \)</span></li></ul></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 of the following are true?</p><p>Select <strong>ALL </strong>correct answers</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\int _{ a }^{ b }{ lnx²dx=2\int _{ a }^{ b }{ lnxdx } } \)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\int _{0 }^{ \pi }{ sinxdx=\int _{ \pi }^{ 0 }{ \frac { 1 }{ sinx } dx } } \)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\int _{ 0 }^{ 2 }{ [x+f(x)]dx } =2+\int _{ 0 }^{ 2 }{ f(x)dx } \)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\int _{ 3 }^{ 5 }{ f(x)dx } =\int _{ 2 }^{ 5 }{ f(x)dx } -\int _{ 2 }^{ 3 }{ f(x)dx } \)</span></label></p></div><div class="q-explanation"><p>Note that lnx²=2lnx</p><p><span class="math-tex">\(\int _{ 0 }^{ 2 }{ [x+f(x)]dx } \\ =\int _{ 0 }^{ 2 }{ xdx } +\int _{ 0 }^{ 2 }{ f(x)dx } \\ ={ \left[ \frac { x² }{ 2 } \right] }_{ 0 }^{ 2 }+\int _{ 0 }^{ 2 }{ f(x)dx } \\ =2+\int _{ 0 }^{ 2 }{ f(x)dx } \)</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 <strong><em>f</em></strong></p><p><strong><em><img alt="" src="../../files/integration/mixed-integration/quiz2/q9" style="width: 400px; height: 285px;"></em></strong></p><p>Evaluate the following definite integral</p></div><div class="q-answer"><p><span class="math-tex">\(\int _{ 3 }^{ -1 }{ f(x)dx } \)</span> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\int _{ -1 }^{ 3 }{ f(x)dx } \)</span> represents the following definite integral</p><p><img alt="" src="../../files/integration/mixed-integration/quiz2/q9explanation.png" style="width: 300px; height: 178px;"></p><p>Therefore, <span class="math-tex">\(\int _{ -1 }^{ 3 }{ f(x)dx } =2\)</span></p><p>So, <span class="math-tex">\(\int _{ 3}^{ -1 }{ f(x)dx } =-2\)</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>The graph shows the function <strong><em>f</em></strong></p><p><img alt="" src="../../files/integration/mixed-integration/quiz2/q10" style="width: 500px; height: 230px;"></p><p><strong><em><span class="math-tex">\(\int _{ -1 }^{ a }{ f(x)dx } =4.5\)</span></em></strong></p><p>Find the value of<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="5"> <span class="review"></span></p></div><div class="q-explanation"><p>The graph below shows that the definite integral <strong><em><span class="math-tex">\(\int _{ -1 }^{ 5 }{ f(x)dx } =4.5\)</span></em></strong></p><p><img alt="" src="../../files/integration/mixed-integration/quiz2/q10explanation.png" style="width: 400px; height: 179px;"></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 a quiz that practises <em><strong>Areas between Graphs</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#c36f8cce"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="c36f8cce"> <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%;"> Mixed SL Integration 4 <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="11-395-1165" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The two functions, <strong><em>f</em></strong> and <strong><em>g</em></strong> are</p><p style="margin-left: 40px;">f(x) = cos²x</p><p style="margin-left: 40px;">g(x) = cos2x</p><p>Which one of the following will give the area of the region shaded below</p><p><img alt="" src="../../files/integration/mixed-integration/quiz-4/q2" style="width: 400px; height: 285px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\int _{ 0 }^{ \pi }{ g(x)dx } -\int _{ 0 }^{ \pi }{ f(x)dx } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\int _{ \pi }^{ 0 }{ f(x)dx } -\int _{ \pi }^{ 0 }{ g(x)dx } \)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\int _{ 0 }^{ \pi }{ \left[ f(x)-g(x) \right] dx } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\int _{ \pi }^{ 0 }{ \left[ f(x)-g(x) \right] dx } \)</span></label></p></div><div class="q-explanation"><p>f(x) = cos²x is the upper curve</p><p>We need to find the area under <strong><em>f</em></strong> and then subtract the area under <strong><em>g</em></strong></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>Let <span class="math-tex">\(f(x)=e^{x-1}\)</span></p><p>The diagram below shows the graph of y = f(x) and the tangent at the point x = 1</p><p>Which of the following gives the area of the shaded region</p><p><img alt="" src="../../files/integration/mixed-integration/quiz-4/q3" style="width: 400px; height: 459px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(1-\int _{ 0 }^{ 1 }{ f(x)dx } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\int _{ 0 }^{ 1 }{ f(x)dx } -1\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(0.5-\int _{ 0 }^{ 1 }{ f(x)dx } \)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\int _{ 0 }^{ 1 }{ f(x)dx } -0.5\)</span></label></p></div><div class="q-explanation"><p>We need to find the area under the curve and subtract the area under the tangent (the triangle).</p><p>The area under the curve = <span class="math-tex">\(\int _{ 0 }^{ 1 }{ f(x)dx }\)</span></p><p>The height of the triangle is <span class="math-tex">\(f(1)=e^0=1\)</span></p><p>The area of the triangle = <span class="math-tex">\(\frac{1\times1}{2}=0.5\)</span></p><p>Hence the shaded area = <span class="math-tex">\(\int _{ 0 }^{ 1 }{ f(x)dx } -0.5\)</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 area of the region bounded by the graph <span class="math-tex">\(y = \frac{1}{x}\)</span>, x = 1 , x = e and the x axis</p></div><div class="q-answer"><p>Area = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/integration/mixed-integration/quiz-4/q5" style="width: 300px; height: 286px;"></p><p>The area of the region can be found by working out the following integral</p><p><span class="math-tex">\(\int _{ 1 }^{ e }{ \frac { 1 }{ x } dx={ [ln|x|] }_{ 1 }^{ e } } \)</span></p><p>= lne - ln1</p><p>= 1 - 0</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 class="exercise shadow-bottom"><div class="q-question"><p>The area of the region bounded by the graph <span class="math-tex">\(y = \frac{1}{2x}\)</span>, x = <strong><em>a</em></strong>, x = 1 and the x axis is equal to <span class="math-tex">\(ln3\)</span></p><p>Work out the value of <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="9"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\int _{ 1 }^{ a }{ \frac { 1 }{ 2x } dx } \)</span></p><p><span class="math-tex">\(=\frac { 1 }{ 2 } \int _{ 1 }^{ a }{ \frac { 1 }{ x } dx } \)</span></p><p><span class="math-tex">\(=\frac { 1 }{ 2 } { [ln|x|] }_{ 1 }^{ a }\)</span></p><p><span class="math-tex">\(=\frac { 1 }{ 2 } (lna-ln1)\)</span></p><p><span class="math-tex">\(=\frac { 1 }{ 2 } lna\)</span></p><p><span class="math-tex">\(=ln{ a }^{ \frac { 1 }{ 2 } }\)</span></p><hr class="hidden-separator"><span class="math-tex">\(ln{ a }^{ \frac { 1 }{ 2 } }=ln3 \\ { a }^{ \frac { 1 }{ 2 } }=3 \\a=9\)</span></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"> The diagram below shows the graphs of y = x² and y = 2 - x²</p><p>The area of the shaded region = <span class="math-tex">\(\frac{a}{3}\)</span></p><p>Find the value of <strong><em>a</em></strong></p><p><img alt="" src="../../files/integration/mixed-integration/quiz-4/q7" style="width: 400px; height: 340px;"></p></div><div class="q-answer"><p><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p>Area = <span class="math-tex">\(\int _{ -1 }^{ 1 }{ (2-x²)dx } -\int _{ -1 }^{ 1 }{ x²dx } \)</span></p><p>We can put the two integrals together</p><p>Area = <span class="math-tex">\(\int _{ -1 }^{ 1 }{ (2-x²-x²)dx } \)</span></p><p><span class="math-tex">\(=\int _{ -1 }^{ 1 }{ (2-2x²)dx } \)</span></p><p><span class="math-tex">\(={ \left[ 2x-\frac { 2{ x }^{ 3 } }{ 3 } \right] }_{ -1 }^{ 1 }\)</span></p><p><span class="math-tex">\(=\left[ 2-\frac { 2 }{ 3 } \right] -\left[ -2+\frac { 2 }{ 3 } \right] \)</span></p><p><span class="math-tex">\(=\frac{8}{3}\)</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 diagram below shows the graphs of y = sinx and y = cosx.</p><p>The area of the shaded region = <span class="math-tex">\(a\sqrt{2}\)</span></p><p>Find the value of <strong><em>a</em></strong></p><p><img alt="" src="../../files/integration/mixed-integration/quiz-4/q8" style="width: 400px; height: 295px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>Area = <span class="math-tex">\(\int _{ \frac { \pi }{ 4 } }^{ \frac { 5\pi }{ 4 } }{ sinxdx } -\int _{ \frac { \pi }{ 4 } }^{ \frac { 5\pi }{ 4 } }{ cosxdx } \)</span></p><p>We can put the two integrals together</p><p>Area = <span class="math-tex">\(\int _{ \frac { \pi }{ 4 } }^{ \frac { 5\pi }{ 4 } }{ (sinx-cosx)dx } \)</span></p><p><span class="math-tex">\(={ \left[ -cosx-sinx \right] }_{ \frac { \pi }{ 4 } }^{ \frac { 5\pi }{ 4 } }\)</span></p><p><span class="math-tex">\(=\left[ -cos\frac { 5\pi }{ 4 } -sin\frac { 5\pi }{ 4 } \right] -\left[ -cos\frac { \pi }{ 4 } -sin\frac { \pi }{ 4 } \right] \)</span></p><p><span class="math-tex">\(=\left[ \frac { \sqrt { 2 } }{ 2 } +\frac { \sqrt { 2 } }{ 2 } \right] -\left[ -\frac { \sqrt { 2 } }{ 2 } -\frac { \sqrt { 2 } }{ 2 } \right] \)</span></p><p><span class="math-tex">\(=\left[ \sqrt { 2 } \right] -\left[ -\sqrt { 2 } \right] \)</span></p><p><span class="math-tex">\(=2\sqrt { 2 } \)</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"> Let f(x) = sinx and g(x) = sin2x</p><p>The graphs of y = f(x) and y = g(x) meet at x = 0 and x = <span class="math-tex">\(\frac{\pi}{3}\)</span> as shown below.</p><p>Find the area of the shaded region</p><p><img alt="" src="../../files/integration/mixed-integration/quiz-4/q9" style="width: 400px; height: 295px;"></p></div><div class="q-answer"><p>Area = <input type="text" style="height: auto;" data-c="0.25"> <span class="review"></span></p></div><div class="q-explanation"><p>Area = <span class="math-tex">\(\int _{ 0 }^{ \frac { \pi }{ 3 } }{ (sin2x-sinx)dx } \)</span></p><p><span class="math-tex">\(={ \left[ -\frac { cos2x }{ 2 } +cosx \right] }_{ 0 }^{ \frac { \pi }{ 3 } }\)</span></p><p><span class="math-tex">\(=\left[ -\frac { cos\frac { 2\pi }{ 3 } }{ 2 } +cos\frac { \pi }{ 3 } \right] -\left[ -\frac { cos0 }{ 2 } +cos0 \right] \)</span></p><p><span class="math-tex">\(=\left[ \frac { 1 }{ 4 } +\frac { 1 }{ 2 } \right] -\left[ -\frac { 1 }{ 2 } +1 \right] \)</span></p><p><span class="math-tex">\(=\frac { 1 }{ 4 } \)</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>The diagram below shows the graph of <span class="math-tex">\(y = \frac{2x}{x^2-1}\)</span></p><p>The shaded region has an area <span class="math-tex">\(lna\)</span></p><p>Find the value of <strong><em>a</em></strong></p><p><img alt="" src="../../files/integration/mixed-integration/quiz-4/q10" style="width: 400px; height: 352px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>Area = <span class="math-tex">\(\int _{ 2 }^{ 4 }{ \frac { 2x }{ x²-1 } dx } \)</span></p><p>We might recognise that this integral is in the form <span class="math-tex">\(=\int { \frac { f'(x) }{ f(x) } dx=ln|f(x)|+C } \)</span></p><p>Or, we could use Integration by substitution with the substitution u = x² - 1</p><p><span class="math-tex">\(\int _{ 2 }^{ 4 }{ \frac { 2x }{ x²-1 } dx } ={ \left[ ln|x²-1| \right] }_{ 2 }^{ 4 }\)</span></p><p>= ln 15 - ln 3</p><p><span class="math-tex">\(=ln\frac { 15 }{ 3 } \)</span></p><p>= ln3</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 a quiz that practises <em><strong>Kinematics</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#4bf9fbc2"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="4bf9fbc2"> <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%;"> Mixed SL Integration 5 <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="11-396-1165" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>A rocket starts from rest and accelerates such that <span class="math-tex">\(a(t)=3t+t^2\)</span>, where <em><strong>t</strong></em> = time</p><p>Find a formula for the velocity of the rocket at time <strong><em>t</em></strong></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { 3t^{ 2 } }{ 2 } +\frac { { t }^{ 3 } }{ 3 } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 3t^{ 2 } }{ 2 } +\frac { { t }^{ 3 } }{ 3 } +C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(3+2t\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(3+t\)</span></label></p></div><div class="q-explanation"><p>In order to find velocity, we need to integrate the acceleration formula</p><p><span class="math-tex">\(v(t)=\int { (3t+t^{ 2 })dt } \\ =\frac { 3t^{ 2 } }{ 2 } +\frac { { t }^{ 3 } }{ 3 } +C\)</span></p><p>The rocket starts from rest, so when t = 0, v = 0</p><p><span class="math-tex">\(0=\frac { 3(0^{ 2 }) }{ 2 } +\frac { { 0 }^{ 3 } }{ 3 } +C\)</span></p><p>C = 0</p><p><span class="math-tex">\(v(t) =\frac { 3t^{ 2 } }{ 2 } +\frac { { t }^{ 3 } }{ 3 } \)</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"> The velocity of an object is given by <span class="math-tex">\(v(t)=-2sint\)</span></p><p>Find the displacement of the object relative to its initial position at <span class="math-tex">\(t=\frac{3\pi}{2}\)</span></p></div><div class="q-answer"><p>displacement = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p>To find displacement, we find the definite integral of velocity</p><p><span class="math-tex">\(s(t)=\int _{ 0 }^{ \frac { 3\pi }{ 2 } }{ -2sintdt } \)</span></p><p><span class="math-tex">\(={ \left[ 2cost \right] }_{ 0 }^{ \frac { 3\pi }{ 2 } }\)</span></p><p><span class="math-tex">\(=\left[ 2cos\left( \frac { 3\pi }{ 2 } \right) \right] -\left[ 2cos0 \right] \)</span></p><p>= 0 - 2</p><p>= -2</p><p>The object has travelled backwards and is behind its initial position.</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"> In this question, distances are given in metres and time, t in seconds</p><p>The velocity-time graph below shows the velocity of an object in the first 3 seconds.</p><p><span class="math-tex">\(v(t) = 2tsin(t^2)\)</span></p><p>Find the <strong><em>distance travelled</em></strong> in the first <span class="math-tex">\(\sqrt{2\pi}\)</span> seconds</p><p><img alt="" src="../../files/integration/mixed-integration/quiz5/q3" style="width: 400px; height: 354px;"></p></div><div class="q-answer"><p>Distance travelled = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> m</p></div><div class="q-explanation"><p>In order to integrate the function, we can use integration by substitution with the substitution u = t²</p><table border="0" cellpadding="0" cellspacing="0" height="67" width="698"><tbody><tr><td><span class="math-tex">\(\int { 2tsin(t²)dt } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u=t²\\ \frac { du }{ dt } =2t\\ du=2tdt\)</span></td></tr><tr><td><span class="math-tex">\(\int { sinudu } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=-cosu+C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=-cos(t²)+C\)</span></td><td> </td></tr></tbody></table><hr class="hidden-separator"><p>You might understand from the graph that the object travels backwards first.</p><p>If we find the definite integral from 0 to <span class="math-tex">\(\sqrt{2\pi}\)</span>, this will find the displacement of the object relative from the starting position. The WRONG answer!</p><p>The distance travelled is given by the area under the graph.</p><p>We need to split the area into 2 parts</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int _{ 0 }^{ \sqrt { \pi } }{ 2tsin(t²)dt } \\ =\int _{ 0 }^{ \pi }{ sinudu } \\ ={ \left[ -cosu \right] }_{ 0 }^{ \pi }\\ =\left[ -cos(\pi ) \right] -\left[ -cos(0) \right] \\ =1+1\\ =2\)</span></td><td><span class="math-tex">\(\int _{ \sqrt { \pi } }^{ \sqrt { 2\pi } }{ 2tsin(t²)dt } \\=\int _{ \pi }^{ 2\pi }{ sinudu } \\={ \left[ -cosu \right] }_{ \pi }^{ 2\pi }\\=\left[ -cos(2\pi ) \right] -\left[ -cos(\pi ) \right]\\ =-1-1\\=-2\)</span></td></tr></tbody></table><p>Distance travelled in the first part = 2</p><p>Distance travelled in the second part = 2</p><p>Total distance = 4</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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> A object moves along a straight line. The velocity <em>v</em> m s<sup>−1</sup> after <em>t</em> seconds is given by</p><p><span class="math-tex">\(v(t)=3sint-t^{cost}\)</span>, for 0 ≤ <em>t</em> ≤ 6</p><p>Find the displacement after 6 seconds to 3 significiant figures</p><p><img alt="" src="../../files/integration/mixed-integration/quiz5/q4" style="width: 400px; height: 314px;"></p></div><div class="q-answer"><p>Displacement = <input type="text" style="height: auto;" data-c="-6.47"> <span class="review"></span></p></div><div class="q-explanation"><p>Use your calculator to find the definite integral <span class="math-tex">\(\int _{ 0 }^{ 6 }{ (3sint-t^{ cost })dt } \)</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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> A object moves along a straight line. The velocity <em>v</em> m s<sup>−1</sup> after <em>t</em> seconds is given by</p><p><span class="math-tex">\(v(t)=\sqrt{t}cos(\frac{πt}{4})\)</span></p><p>Find the total distance travelled in the first 6 seconds.</p><p><img alt="" src="../../files/integration/mixed-integration/quiz5/q5" style="width: 400px; height: 280px;"></p></div><div class="q-answer"><p>Total distance travelled = <input type="text" style="height: auto;" data-c="6.07"> <span class="review"></span> </p></div><div class="q-explanation"><p>You might understand from the graph that the object travels forwards first then backwards.</p><p>If we find the definite integral from 0 to 6, this will find the displacement of the object relative from the starting position. The WRONG answer!</p><p>The distance travelled is given by the area under the graph.</p><p>Use your graphical calculator to find the total area</p><hr class="hidden-separator"><p><span class="math-tex">\(\int _{ 0 }^{ 2 }{ \sqrt { t } cos(\frac { πt }{ 4 } )dt } \approx 1.011\\ \int _{ 2 }^{ 6 }{ \sqrt { t } cos(\frac { πt }{ 4 } )dt } \approx -5.062\)</span></p><p>Area <span class="math-tex">\(\approx 1.01+5.06\approx6.07\)</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"> <div class="filter-hl-only"> <p>Here's a quiz that practises <em><strong>Volumes of Revolution</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#3ed0fd62"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="3ed0fd62"> <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%;"> Mixed SL Integration 6 <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="11-397-1165" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The shaded region is rotated 360° about the x axis.</p><p>Which of the following will find the volume of the solid formed?</p><p><img alt="" src="../../files/integration/mixed-integration/quiz-6/q1" style="width: 400px; height: 305px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\pi \int _{ 1 }^{ \pi }{ { g(x) }dx } -\pi \int _{ 1 }^{ \pi }{ { f(x) }dx } \)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\pi \int _{ 1 }^{ \pi }{ { ([g(x)] }^{ 2 }-{ \left[ f(x) \right] }^{ 2 })dx } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\pi \int _{ 1 }^{ \pi }{ { [g(x)-f(x)] }^{ 2 }dx } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\pi \int _{ 1 }^{ \pi }{( { \left[ f(x) \right] }^{ 2 }-{ [g(x)] }^{ 2 })dx } \)</span></label></p></div><div class="q-explanation"><p>Find</p><p style="margin-left: 40px;">the volume formed by <strong><em>g</em></strong></p><p style="margin-left: 40px;">then subtract the volume formed by <strong><em>f</em></strong></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>Let f(x) =lnx and g(x) = 2ln(x - 2)</p><p>The graphs of <strong><em>f </em></strong>and <em><strong>g</strong></em><strong><em> </em></strong>are shown below.</p><p>The shaded region is rotated 360° about the x axis.</p><p>Which of the following will find the volume of the solid formed?</p><p><img alt="" src="../../files/integration/mixed-integration/quiz-6/q2" style="width: 300px; height: 284px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\pi \int _{ 1 }^{ 4 }{ { (\left[ lnx \right] }^{ 2 }-{ \left[ 2ln(x-2) \right] }^{ 2 })dx } \)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\pi \int _{ 1 }^{ 4 }{ { \left[ lnx \right] }^{ 2 }dx } - \pi \int _{ 3 }^{ 4 }{ { \left[ 2ln(x-2) \right] }^{ 2 }dx } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\pi \int _{ 3 }^{ 4 }{ { \left[ 2ln(x-2) \right] }^{ 2 }dx } -\pi \int _{ 1 }^{ 4 }{ { \left[ lnx \right] }^{ 2 }dx } \quad \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\pi \int _{ 1 }^{ 3 }{ { \left[ lnx \right] }^{ 2 }dx } - \pi \int _{ 3 }^{ 4 }{ { \left[ 2ln(x-2) \right] }^{ 2 }dx } \)</span></label></p></div><div class="q-explanation"><p>The graph in black is f(x) =lnx</p><p>The graph in blue is g(x) = 2ln(x - 2)</p><p>Consider a region under each function</p><p><img alt="" src="../../files/integration/mixed-integration/quiz-6/q2expl.png" style="width: 650px; height: 192px;"></p><p>We can see that we need to work out <span class="math-tex">\(\pi \int _{ 1 }^{ 4 }{ { \left[ f(x) \right] }^{ 2 }dx } -\pi \int _{ 3 }^{ 4 }{ { \left[ g(x) \right] }^{ 2 }dx } \)</span></p><p><span class="math-tex">\(=\pi \int _{ 1 }^{ 4 }{ { \left[ lnx \right] }^{ 2 }dx } - \pi \int _{ 3 }^{ 4 }{ { \left[ 2ln(x-2) \right] }^{ 2 }dx } \)</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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Let <span class="math-tex">\(f(x)=(x+1)(x-1)(x-3)\)</span></p><p>The region enclosed by the graph of <strong><em>f</em></strong> and the x axis is rotated 360° around the x axis.</p><p>The volume of the the solid formed is <span class="math-tex">\(\frac{a}{35}\pi\)</span>.</p><p>Find the value of <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="162"> <span class="review"></span></p></div><div class="q-explanation"><p>You can use your graphical calculator for this question.</p><p>If we were calculating the area, we would have to split this into 2 separate integrals, but this is not necessary for volumes (this is no negative value)</p><p><img alt="" src="../../files/integration/mixed-integration/quiz-6/q3explanation" style="width: 300px; height: 239px;"></p><p>Volume = <span class="math-tex">\(\pi\int _{ -1 }^{ 3 }{ { \left[ (f(x) \right] }^{ 2 }dx } \)</span></p><p><span class="math-tex">\( =\pi\int _{ -1 }^{ 3 }{ { \left[ (x+1)(x-1)(x-3) \right] }^{ 2 }dx } \\ =\frac { 162 }{ 35 } \pi \\ \approx 14.5\)</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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Let <span class="math-tex">\(f(x) = 3xln(2-x^2)\)</span></p><p>The graph of <strong><em>f</em></strong> has a minimum value at x = <strong><em>a</em></strong></p><p>The shaded region is rotated 360° around the x axis. Find the volume of the solid formed.</p><p>Give your answer to 3 significant figures.</p><p><img alt="" src="../../files/integration/mixed-integration/quiz-6/q4" style="width: 400px; height: 357px;"></p></div><div class="q-answer"><p>Volume <span class="math-tex">\(\approx\)</span> <input type="text" style="height: auto;" data-c="0.644"> <span class="review"></span></p></div><div class="q-explanation"><p>This is a question for your graphical calculator.</p><p>We can find the x coordinate of A (minimum point). It is best to find this to 4 significant figures.</p><p style="margin-left: 40px;"><em><strong>-0.6214</strong></em></p><p>We also need to find the x coordinate of B (x intercept)</p><p style="margin-left: 40px;"><em><strong>1</strong></em></p><p>The volume is found by calculating <span class="math-tex">\(\pi\int _{ -0.6214}^{ 1 }{ { \left[ f(x) \right] }^{ 2 }dx } \)</span></p><p><em><strong>Don't forget <span class="math-tex">\(\pi\)</span></strong></em></p><p>On your calculator, work out <span class="math-tex">\(\pi\int _{ -0.6214 }^{ 1 }{ { \left[ 3xln(2-x^{ 2 }) \right] }^{ 2 }dx } \)</span> and round it to 3 significant figures</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"> The region <strong><em>R</em></strong> is bounded by the curve <span class="math-tex">\(y=\frac{4}{x}\)</span> and the lines y = 0, x = 1 and x = 2</p><p>The volume of the solid generated by rotatring <strong><em>R</em></strong> 360° around the x axis is given by <span class="math-tex">\(a\pi\)</span>.</p><p>Find the value of <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="8"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/integration/mixed-integration/quiz-6/q5explanation" style="width: 300px; height: 253px;"></p><p>Volume = <span class="math-tex">\(\pi \int _{ 1 }^{ 2 }{ { \left[ (f(x) \right] }^{ 2 }dx } \)</span></p><p><span class="math-tex">\(=\pi \int _{ 1 }^{ 2 }{ { \left[ \frac { 4 }{ x } \right] }^{ 2 }dx } \)</span></p><p><span class="math-tex">\(=\pi \int _{ 1 }^{ 2 }{ { 16 }{ x }^{ -2 }dx } \)</span></p><p><span class="math-tex">\(=\pi { \left[ -16{ x }^{ -1 } \right] }_{ 1 }^{ 2 }\)</span></p><p><span class="math-tex">\(=\pi \left[ -\frac { 16 }{ x } \right] _{ 1 }^{ 2 }\)</span></p><p><span class="math-tex">\(=\pi [-8+16]\)</span></p><p><span class="math-tex">\(=8\pi \)</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"> The diagram shows the graph of the function <span class="math-tex">\(f(x) = \sqrt{2x-2}\)</span> and the normal at the point where x = 3 with equation y = -2x + 8</p><p>The shaded region is rotated 360° around the x axis and the volume of the solid created is <span class="math-tex">\(\frac{a\pi}{3}\)</span> </p><p>Work out the value of <strong><em>a</em></strong></p><p><img alt="" src="../../files/integration/mixed-integration/quiz-6/q6" style="width: 400px; height: 305px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="16"> <span class="review"></span></p></div><div class="q-explanation"><p>The solid is made up of 2 parts</p><ol><li>the region formed by the curve</li><li>the cone formed by the straight line</li></ol><p><img alt="" src="../../files/integration/mixed-integration/quiz-6/q6expl.png" style="width: 650px; height: 198px;"></p><p>Volume = <span class="math-tex">\(\pi \int _{ 1 }^{ 3 }{ { (2x-2) }dx } +\frac { 1 }{ 3 } \pi \cdot { 2 }^{ 2 }\cdot 1\)</span></p><p><span class="math-tex">\(=\pi { \left[ x^{ 2 }-2x \right] }_{ 1 }^{ 3 }+\frac { 4\pi }{ 3 } \)</span></p><p><span class="math-tex">\(=\pi { \left[ 9-6 \right] }-\pi { \left[ 1-2 \right] }+\frac { 4\pi }{ 3 } \)</span></p><p><span class="math-tex">\(=4\pi +\frac { 4\pi }{ 3 } \)</span></p><p><span class="math-tex">\(=\frac { 16\pi }{ 3 } \)</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> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-green panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself further</p> </div> </div> <div class="panel-body"> <div> <p>You can get further practice by trying a dynamic quiz.</p> <p>If you refresh this page, you will get a new set of quizzes</p> <div class="panel panel-green 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><img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> Medium</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz dyn-quiz" data-pid="1165" data-mtime="1675506255.9063"><div class="label label-default q-number">1</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="actions"><span class="score" data-score="0"></span><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="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows the function f</p><p>Evaluate the following definite integral</p><p><img alt="" src="../../files/integration/definite-integral-and-area/q5.png" style="width: 250px; height: 233px;"></p></div><div class="q-answer"><p><span class="math-tex">\(\int _{ -1 }^{ 0 }{ f(x)dx } \)</span> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p><hr class="hidden-separator"></div><div class="q-explanation"><p><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</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">3</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"><span class="math-tex">\(a_1\)</span> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></div><div class="q-answer"><span class="math-tex">\(a_2\)</span> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></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="actions"><span class="score" data-score="0"></span><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="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>A rocket starts from rest and accelerates such that <span class="math-tex">\(a(t)=3t+t^2\)</span>, where <em><strong>t</strong></em> = time</p><p>Find a formula for the velocity of the rocket at time <strong><em>t</em></strong></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { 3t^{ 2 } }{ 2 } +\frac { { t }^{ 3 } }{ 3 } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(3+2t\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(3+t\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 3t^{ 2 } }{ 2 } +\frac { { t }^{ 3 } }{ 3 } +C\)</span></label></p></div><div class="q-explanation"><p>In order to find velocity, we need to integrate the acceleration formula</p><p><span class="math-tex">\(v(t)=\int { (3t+t^{ 2 })dt } \\ =\frac { 3t^{ 2 } }{ 2 } +\frac { { t }^{ 3 } }{ 3 } +C\)</span></p><p>The rocket starts from rest, so when t = 0, v = 0</p><p><span class="math-tex">\(0=\frac { 3(0^{ 2 }) }{ 2 } +\frac { { 0 }^{ 3 } }{ 3 } +C\)</span></p><p>C = 0</p><p><span class="math-tex">\(v(t) =\frac { 3t^{ 2 } }{ 2 } +\frac { { t }^{ 3 } }{ 3 } \)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int\frac{2x}{x^2+1} \mathrm{d}x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(2\ln |x^2+1|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{x^2+1}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{2}\ln|x^2+1|+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\ln |x^2+1|+C\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{\mathrm{d}}{\mathrm{d}x}(x^2+1)=2x\)</span></p><p>Hence integral is in the form <span class="math-tex">\(\int\frac{f'(x)}{f(x)} \mathrm{d}x=\ln |f(x)|+C\)</span></p><p><span class="math-tex">\(\int\frac{2x}{x^2+1} \mathrm{d}x=\ln |x^2+1|+C\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-green 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><img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> Hard</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz dyn-quiz" data-pid="1165" data-mtime="1675506255.9093"><div class="label label-default q-number">1</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="actions"><span class="score" data-score="0"></span><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="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\int { \frac { lnx }{ x } dx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { lnx }{ x^2 }+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\((ln|x|)^2+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { { \left( lnx \right) }^{ 2 } }{ 2 } +C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(ln|x|+C\)</span></label></p></div><div class="q-explanation"><p>We can use integration by substitution. Let <span class="math-tex">\(u=lnx\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\int { \frac { lnx }{ x } dx } \)</span></td><td> </td></tr><tr><td> </td><td><span class="math-tex">\(u=lnx\\ \frac { du }{ dx } =\frac { 1 }{ x } \\ du=\frac { 1 }{ x } dx\)</span></td></tr><tr><td><span class="math-tex">\(\int { udu } \)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { { u }^{ 2 } }{ 2 } +C\)</span></td><td> </td></tr><tr><td><span class="math-tex">\(=\frac { { \left( lnx \right) }^{ 2 } }{ 2 } +C\)</span></td><td> </td></tr></tbody></table></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below shows the graph of <span class="math-tex">\(y = \frac{2x}{x^2-1}\)</span></p><p>The shaded region has an area <span class="math-tex">\(lna\)</span></p><p>Find the value of <strong><em>a</em></strong></p><p><img alt="" src="../../files/integration/mixed-integration/quiz-4/q10" style="width: 400px; height: 352px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>Area = <span class="math-tex">\(\int _{ 2 }^{ 4 }{ \frac { 2x }{ x²-1 } dx } \)</span></p><p>We might recognise that this integral is in the form <span class="math-tex">\(=\int { \frac { f'(x) }{ f(x) } dx=ln|f(x)|+C } \)</span></p><p>Or, we could use Integration by substitution with the substitution u = x² - 1</p><p><span class="math-tex">\(\int _{ 2 }^{ 4 }{ \frac { 2x }{ x²-1 } dx } ={ \left[ ln|x²-1| \right] }_{ 2 }^{ 4 }\)</span></p><p>= ln 15 - ln 3</p><p><span class="math-tex">\(=ln\frac { 15 }{ 3 } \)</span></p><p>= ln3</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int\frac{1}{x\ln x} \mathrm{d}x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{x^2(\ln x)^2}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\ln|\ln x|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln|x\ln x|+C\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\int\frac{1}{x\ln x} \mathrm{d}x=\int\frac{\frac{1}{x}}{\ln x} \mathrm{d}x\)</span></p><p><span class="math-tex">\(\frac{\mathrm{d}}{\mathrm{d}x}(\ln x)=\frac{1}{x}\)</span></p><p>Hence integral is in the form <span class="math-tex">\(\int\frac{f'(x)}{f(x)} \mathrm{d}x=\ln |f(x)|+C\)</span></p><p><span class="math-tex">\(\int\frac{\frac{1}{x}}{\ln x} \mathrm{d}x=\ln|\ln x|+C\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large \int_0^{1} \frac {x(e^{x^2}+1)}{e^{x^2}+x^2} {d x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{1}{(e+1)^2}-1\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large\frac{1}{2}\ln(e+1)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\ln(e+1)^2\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\ln(e+1)\)</span></label></p></div><div class="q-explanation"><p>This is an integration by recognition question in the form <span class="math-tex">\(\large \int \frac {f'(x)}{f(x)} {d x}=\ln |f(x)|+C\)</span></p><p>We can also use integration by substitution using the substitution <span class="math-tex">\(\large u=\cos x\)</span></p><p>Note that, <span class="math-tex">\(\large \frac{d}{dx}(e^{x^2}+x^2)=2xe^{x^2}+2x=2x(e^{x^2}+1)\)</span></p><p>So, we have</p><p><span class="math-tex">\(\large \frac{1}{2}\int_0^{1} \frac {2x(e^{x^2}+1)}{e^{x^2}+x^2} {d x}=\frac{1}{2}[\ln |e^{x^2}+x^2|]_0^{1}\\ \large =\frac{1}{2}\ln([e^1+1]-[e^0+0]) \\ \large=\frac{1}{2}\ln\frac{e+1}{1}\\ \large=\frac{1}{2}\ln(e+1)\)</span></p></div><div class="actions"><span 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