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<p>For $0 < b < 1$, evaluate the following integral in terms of $b$:</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\displaystyle\int\nolimits_{0}^{1}{x}\left| x-b \right|\,\mathrm{d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int\nolimits_{-1}^{0}{x}\left| x-b \right|\,\mathrm{d}x$</div></li></ol>

For $0 < b < 1$, evaluate the following integral in terms of $b$:

Techniques of  Integration - General

ACJC TUTORIAL 22 Q9

Techniques of  Integration

<p>A curve $C$ has equation $y={{\mathrm{e}}^{x}}$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact volume of revolution when the region bounded by $C$, the line $y=3$ and the $y$-axis is rotated completely about the $y$-axis.</span></div><p style="text-align: right;">[5]</p></li><li><div class="list-item-content"><span class="list-item-text">Describe a pair of transformations which transforms the curve with equation $y={{\mathrm{e}}^{1+2x}}$ on to the curve $C$.</span></div><p style="text-align: right;">[2]</p></li></ol>

A curve $C$ has equation $y={{\mathrm{e}}^{x}}$.

Find the exact volume of revolution when the region bounded by $C$, the line $y=3$ and the $y$-axis is rotated completely about the $y$-axis.

Describe a pair of transformations which transforms the curve with equation $y={{\mathrm{e}}^{1+2x}}$ on to the curve $C$.

2019 NJC Promo Q4

<ol class="abc-list"><li><div class="list-item-paragraph">Find $\displaystyle\displaystyle \displaystyle\int{3t{{\tan }^{-1}}\left( 3t \right)\mathrm{d}t}$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Using the substitution $u={{x}^{2}}+1$, show that&nbsp;$\displaystyle\displaystyle \displaystyle\int\nolimits_{0}^{\sqrt{7}}{{{x}^{3}}{{\left( {{x}^{2}}+1 \right)}^{\frac{1}{3}}}\mathrm{d}x}$ can be expressed as&nbsp;$\frac{1}{2}\displaystyle\displaystyle \displaystyle\int\nolimits_{a}^{b}{{{u}^{\frac{4}{3}}}-{{u}^{\frac{1}{3}}}}\mathrm{ d}u$, where&nbsp;$a$ and $b$ are constants to be determined.</div><div class="list-item-paragraph">Hence find the exact value of $\displaystyle\displaystyle \displaystyle\int\nolimits_{0}^{\sqrt{7}}{{{x}^{3}}}{{\left( {{x}^{2}}+1 \right)}^{\frac{1}{3}}}\mathrm{d}x$.<span class="inline-marks">[5]</span></div></li></ol>

2022 HCI Promo Q8

<p>$\displaystyle\displaystyle \displaystyle\int{{{\left( \frac{\sqrt{x}}{{{x}^{2}}+1} \right)}^{2}}}\mathrm{ d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{{{\left( \frac{\sqrt{x}}{{{x}^{2}}+1} \right)}^{2}}}\mathrm{ d}x$

<p>The curve $C$ has parametric equations $x=6{{t}^{2}}$, $y=\frac{2t}{\sqrt{1-{{t}^{2}}}}$, $0 < t < 1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">A line is tangent to the curve $C$ at point $A$ and passes through the origin $O$. Show that the line has equation $y=\frac{2}{3}x$.</div><p style="text-align: right;">[4]</p></li></ol>
<p>The region $R$ is bounded by the curve and the tangent line in <strong>(i)</strong>.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the area of $R$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Write down the Cartesian equation of the curve $C$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the exact volume of the solid of revolution generated when $R$ is rotated completely about the $x$-axis, giving your answer in the form $\left( a\ln b-c \right)\pi $, where constants $a$, $b$, $c$ are to be determined.</div><p style="text-align: right;">[4]</p></li></ol>

The curve $C$ has parametric equations $x=6{{t}^{2}}$, $y=\frac{2t}{\sqrt{1-{{t}^{2}}}}$, $0 < t < 1$.

The region $R$ is bounded by the curve and the tangent line in **(i)**.

2019 NYJC P2 Q2

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that $\sqrt{\frac{1-2x}{1+2x}}$ can be written in the form $\frac{\mathrm{f}\left( x \right)}{\sqrt{1-4{{x}^{2}}}}$ where $\mathrm{f}\left( x \right)$ is a polynomial to be determined. Hence, find $\displaystyle\int{\sqrt{\frac{1-2x}{1+2x}}\,\mathrm{d}x}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that $\displaystyle\int\nolimits_{\mathrm{e}}^{{{\mathrm{e}}^{2}}}{\frac{1}{x\ln x}}\mathrm{ d}x=\ln 2$.</span></div><p style="text-align: right;">[3]</p></li></ol>

Show that $\sqrt{\frac{1-2x}{1+2x}}$ can be written in the form $\frac{\mathrm{f}\left( x \right)}{\sqrt{1-4{{x}^{2}}}}$ where $\mathrm{f}\left( x \right)$ is a polynomial to be determined. Hence, find $\displaystyle\displaystyle \displaystyle\int{\sqrt{\frac{1-2x}{1+2x}}\,\mathrm{d}x}$.

Show that $\displaystyle\displaystyle \displaystyle\int\nolimits_{\mathrm{e}}^{{{\mathrm{e}}^{2}}}{\frac{1}{x\ln x}}\mathrm{ d}x=\ln 2$.

2019 ASRJC P1 Q1

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x=a\left( \frac{4}{\pi }t+\sin 2t \right)+2a$, </p>
<p style="text-align: center">$y=a\cos t$,</p>
<p>where $-\frac{\pi }{2}\le t\le \frac{\pi }{2}$ and $a$ is a positive constant.</p>
<ol class="abc-list"></ol>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch $C$. State clearly the coordinates of any points where $C$ meets the $x$-axis.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that the area enclosed by $C$ and the $x$-axis is given by</div><div class="list-item-paragraph" style="text-align: center">${{a}^{2}}\displaystyle\int\nolimits_{{{\theta }_{1}}}^{{{\theta }_{2}}}{\left( \frac{4}{\pi }\cos t+2\cos t\cos 2t \right)\mathrm{d}t}$</div><div class="list-item-paragraph">where ${{\theta }_{1}}$ and ${{\theta }_{2}}$ are values to be stated.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence find, in terms of $a$, the exact area enclosed by $C$ and the $x$-axis.</div><p style="text-align: right;">[3]</p></li></ol>

$x=a\left( \frac{4}{\pi }t+\sin 2t \right)+2a$, 

where $-\frac{\pi }{2}\le t\le \frac{\pi }{2}$ and $a$ is a positive constant.

Applications of Intergration - General

2021 HCI P2 Q1

Applications of Intergration

<ol class="i-list"><li><div class="list-item-paragraph">$\displaystyle\int{4{{x}^{3}}}{{({{x}^{4}}+7)}^{4}}\mathrm{d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int{\frac{x+2}{{{x}^{2}}+4x+6}}\mathrm{d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int{x{{e}^{{{x}^{2}}+1}}}\mathrm{d}x$</div></li></ol>

Worked Example 5

(b) If $0 < a < 1$, find $\displaystyle\displaystyle \displaystyle\int\nolimits_{0}^{1}{\left( a+x \right)\left| a-x \right|\,}\mathrm{d}x$, leaving your answer in terms of $a$.

2019 ASRJC Promo Q5

<p>By considering $x=4\cos \theta $ where&nbsp;$0\le \theta \le \pi $, find $\displaystyle\int{\sqrt{16-{{x}^{2}}}\,\mathrm{d}x}$.</p>

By considering $x=4\cos \theta $ where $0\le \theta \le \pi $, find $\displaystyle\displaystyle \displaystyle\int{\sqrt{16-{{x}^{2}}}\,\mathrm{d}x}$.

<ol class="i-list"><li data-list-item-id="item-1759586751181-znfq1mieo" data-list-item-path="item-1759586751181-znfq1mieo"><div class="list-item-paragraph">Use the substitution $x=m\tan t$ to find $\displaystyle\displaystyle \displaystyle\int{\frac{1}{\sqrt{{{m}^{2}}+{{x}^{2}}}}\mathrm{d}x}$, where&nbsp;$m$ is a positive constant and $0\le t&lt;\frac{\pi }{2}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759586751698-0leqhorh6" data-list-item-path="item-1759586751698-0leqhorh6"><div class="list-item-paragraph">Find $\displaystyle\displaystyle \displaystyle\int{\frac{x}{\sqrt{{{m}^{2}}+{{x}^{2}}}}\mathrm{d}x}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759586752216-ris4aqh6h" data-list-item-path="item-1759586752216-ris4aqh6h"><div class="list-item-paragraph">Hence, find $\displaystyle\displaystyle \displaystyle\int{\frac{x}{\sqrt{{{m}^{2}}+{{x}^{2}}}}{{\tan }^{-1}}\left( \frac{x}{m} \right)}\,\mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Use the substitution $x=m\tan t$ to find $\displaystyle\displaystyle \displaystyle\int{\frac{1}{\sqrt{{{m}^{2}}+{{x}^{2}}}}\mathrm{d}x}$, where $m$ is a positive constant and $0\le t<\frac{\pi }{2}$.

Find $\displaystyle\displaystyle \displaystyle\int{\frac{x}{\sqrt{{{m}^{2}}+{{x}^{2}}}}\mathrm{d}x}$.

Hence, find $\displaystyle\displaystyle \displaystyle\int{\frac{x}{\sqrt{{{m}^{2}}+{{x}^{2}}}}{{\tan }^{-1}}\left( \frac{x}{m} \right)}\,\mathrm{d}x$.

2022 NJC Promo Q6

<p>Evaluate $\displaystyle\int{\cos \left( 3x \right){{\sin }^{3}}\left( 3x \right)\mathrm{d}x}$.</p>

Evaluate $\displaystyle\displaystyle \displaystyle\int{\cos \left( 3x \right){{\sin }^{3}}\left( 3x \right)\mathrm{d}x}$.

RI Tutorial Q3

Introduction to Techniques of Integration

<ol class="abc-list"><li data-list-item-id="item-1760430500657-1ac8nrbb5" data-list-item-path="item-1760430500657-1ac8nrbb5"><div class="list-item-paragraph">Show that&nbsp;$\frac{4{{x}^{2}}-6x+1}{\left( {{x}^{2}}+1 \right)\left( x-2 \right)}$ can be expressed as&nbsp;$\frac{Ax+B}{{{x}^{2}}+1}-\frac{C}{x-2}$, where&nbsp;$A$, $B$ and $C$ are constants to be determined. Hence find $\displaystyle \displaystyle\int{\frac{4{{x}^{2}}-6x+1}{\left( {{x}^{2}}+1 \right)\left( x-2 \right)}\mathrm{d}x}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760430501557-soilq958r" data-list-item-path="item-1760430501557-soilq958r"><div class="list-item-paragraph">Differentiate ${{\mathrm{e}}^{\sqrt{x}}}$ with respect to $x$. Hence, or otherwise, find $\displaystyle \displaystyle\int{\frac{{{\mathrm{e}}^{\sqrt{x}}}}{\sqrt{x}{{\left( {{e}^{\sqrt{x}}}+1 \right)}^{4}}}\mathrm{d}x}$.<span class="inline-marks">[3]</span></div></li></ol>

2022 NYJC Promo Q2

<ol class="abc-list"><li><div class="list-item-paragraph">The curve $C$ is defined by the equations</div><div class="list-item-paragraph" style="text-align: center">$x=2\mathrm{cosec}t$, $y=5\cos t{{\sin }^{3}}t$, for $0 < t\le \frac{\pi }{2}$.</div><div class="list-item-paragraph">The region $R$ is bounded by $C$, the $x$-axis and $x=4$ in the first quadrant. Find the exact area of the region $R$.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">The diagram below shows the curves ${{C}_{1}}$ and ${{C}_{2}}$ with equations $y=2{{\left( x-1 \right)}^{2}}+3$ and $y=3{{x}^{2}}$ respectively. The region in the first quadrant enclosed by the curves and the $y$-axis is denoted by $S$. Find the volume of the solid generated when the region $S$ is rotated through $2\pi $ radians about the $y$-axis, giving your answer correct to $4$ decimal places.</div><p style="text-align: right;">[3]</p><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759815274286-xtwbydc4wmk.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk4MTUyNzQyODYteHR3YnlkYzR3bWsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxNDE2MSwiZXhwIjoxNzg1NzE0NDYxfQ.hSlYscyiYGr0n9m9lA-FbLFyAjsQQzl-jAERma21Yrs" alt="List item image" /></div></li></ol>

2019 ASRJC Promo Q8

<ol class="abc-list"><li><div class="list-item-paragraph">The curve $C$ has parametric equations</div><div class="list-item-paragraph" style="text-align: center">$x=\sqrt{3}\cos t$, $y=3\sin t$, for $-\frac{\pi }{2}\le t\le \frac{\pi }{2}.$</div><ol class="i-list"><li><div class="list-item-paragraph">Sketch $C$, indicating clearly the exact coordinates of the axial intercepts.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the exact value of the area of the region bounded by $C$ and the $y$–axis.</div><p style="text-align: right;">[5]</p></li></ol></li></ol>

2013 VJC P1 Q10

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">State the derivative of $\tan \left( {{x}^{3}} \right)$.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Find $\displaystyle\int{6{{x}^{5}}{{\sec }^{2}}\left( {{x}^{3}} \right)\mathrm{d}x}$.</span></div><p style="text-align: right;">[3]</p></li></ol>

State the derivative of $\tan \left( {{x}^{3}} \right)$.

Find $\displaystyle\displaystyle \displaystyle\int{6{{x}^{5}}{{\sec }^{2}}\left( {{x}^{3}} \right)\mathrm{d}x}$.

2020 YIJC P2 Q1

<ol class="abc-list" start="2"><li><div class="list-item-paragraph">It is given that&nbsp;$\,\mathrm{f}(x)=\frac{1}{x-1}$.</div><ol class="i-list"><li><div class="list-item-paragraph">On separate diagrams, sketch the graphs of $y=\mathrm{f}\left( \left| x \right| \right)$ and $y=\left| \mathrm{f}\left( x \right) \right|$. 	<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Without the use of graphing calculator, find the exact value of the constant $a$, where&nbsp;$0&lt;a&lt;1$, for which  	$\left| \displaystyle\int\nolimits_{\text{  }-\frac{1}{2}\ }^{\text{ }\frac{1}{2}}{\text{f (}\left| x \right|)\ \mathrm{d}x} \right|=\displaystyle\int\nolimits_{\text{  }0\ }^{\text{ }a}{\left| \text{f (}x) \right|\ \mathrm{d}x}$.	<span class="inline-marks">[4]</span></div></li></ol></li></ol>

2013 ACJC P2 Q4 

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that $y=\frac{\ln x}{x}$, find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence, or otherwise, find the exact value of $\displaystyle\int\nolimits_{1}^{\mathrm{e}}{\frac{\ln x}{{{x}^{2}}}\mathrm{d}x}$, showing your working.</span></div><p style="text-align: right;">[4]</p></li></ol>

Given that $y=\frac{\ln x}{x}$, find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$.

Hence, or otherwise, find the exact value of $\displaystyle\displaystyle \displaystyle\int\nolimits_{1}^{\mathrm{e}}{\frac{\ln x}{{{x}^{2}}}\mathrm{d}x}$, showing your working.

N2018 P1 Q1

<ol class="abc-list"><li data-list-item-id="item-1760428069804-2yobrldf6" data-list-item-path="item-1760428069804-2yobrldf6"><div class="list-item-paragraph"><p>Find $\displaystyle\displaystyle \displaystyle\int{{{\cot }^{2}} \; 2x\,}\mathrm{d}x$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760428070502-1huyz3ye5" data-list-item-path="item-1760428070502-1huyz3ye5"><div class="list-item-paragraph">Write down the constants $A$ and $B$ such that, for all values of $x$, $-2x-1=A\left( -2x-2 \right)+B$.</div><div class="list-item-paragraph">Hence, find $\displaystyle\displaystyle \displaystyle\int{\frac{-2x-1}{\sqrt{4-2x-{{x}^{2}}}}\,\mathrm{d}x}$.<span class="inline-marks">[4]</span></div></li></ol>

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