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N(A) E Math

PSLE

PSLE Foundation Mathematics

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Find $\displaystyle\displaystyle \displaystyle\int (2x-1) \cos x \, \mathrm{d}x$.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Hence find the value of $\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi}{2}} |2x-1| \cos x \, \mathrm{d}x$. Give your answer in the form $A-4 \cos B$, where&nbsp;$A$ and $B$ are exact constants to be determined.</span><span class="inline-marks">[4]</span></div></li></ol>

Find $\displaystyle\displaystyle \displaystyle\int (2x-1) \cos x \, \mathrm{d}x$.

Hence find the value of $\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi}{2}} |2x-1| \cos x \, \mathrm{d}x$. Give your answer in the form $A-4 \cos B$, where $A$ and $B$ are exact constants to be determined.

Integrals involving Modulus Functions

Integration by Parts

2025 HCI P1 Q9

Applications of Integration

Techniques of Integration

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-content"><span class="list-item-text">Find $\displaystyle\displaystyle \displaystyle\int_{0}^{4} \mathrm{e}^{|x-1|} \, \mathrm{d}x$, leaving your answer in exact form.</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b"><div class="list-item-content"><span class="list-item-text">Find $\displaystyle\displaystyle \displaystyle\int \sin^{2} 3x + \tan^{2} 3x \, \mathrm{d}x$.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c"><div class="list-item-content"><span class="list-item-text">Use the substitution $x = u^{6}$, where&nbsp;$u &gt; 0$, to find $\displaystyle\displaystyle \displaystyle\int \frac{1}{\sqrt{x} + \sqrt[3]{x}} \, \mathrm{d}x$.</span><span class="inline-marks">[5]</span></div></li></ol>

Find $\displaystyle\displaystyle \displaystyle\int_{0}^{4} \mathrm{e}^{|x-1|} \, \mathrm{d}x$, leaving your answer in exact form.

Find $\displaystyle\displaystyle \displaystyle\int \sin^{2} 3x + \tan^{2} 3x \, \mathrm{d}x$.

Use the substitution $x = u^{6}$, where $u > 0$, to find $\displaystyle\displaystyle \displaystyle\int \frac{1}{\sqrt{x} + \sqrt[3]{x}} \, \mathrm{d}x$.

Integrating Powers of Sine and Cosine

Integrable Forms of Trigonometric Functions

Integrals Involving Long Division

Integration by Substitution

2025 EJC P1 Q10

<p>Given that $a>0$, find $\displaystyle\int_{0}^{3a}{\left| {{x}^{2}}-{{a}^{2}} \right|\mathrm{d}x}$ in terms of $a$.</p>

Given that $a>0$, find $\displaystyle\displaystyle \displaystyle\int_{0}^{3a}{\left| {{x}^{2}}-{{a}^{2}} \right|\mathrm{d}x}$ in terms of $a$.

Worked Example 1

<p>Given that&nbsp;$a &gt; 0$, find $\displaystyle \displaystyle\int_{0}^{3a} |x^{2} - a^{2}|$ $dx$ in terms of $a$.</p>

Given that $a > 0$, find $\displaystyle \displaystyle\int_{0}^{3a} |x^{2} - a^{2}|$ $dx$ in terms of $a$.

<p>Given that&nbsp;$a &gt; 0$, find $\displaystyle \displaystyle\int_{0}^{4a} |(x - 3a)(x - a)|$ $dx$ in terms of $a$.</p>

Given that $a > 0$, find $\displaystyle \displaystyle\int_{0}^{4a} |(x - 3a)(x - a)|$ $dx$ in terms of $a$.

<ol class="i-list"><li data-list-item-id="item-1761369990416-o3xpf920n" data-list-item-path="item-1761369990416-o3xpf920n"><div class="list-item-paragraph">By sketching the graphs of $y=3{{\mathrm{e}}^{x}}$ and $y=x+3$ or otherwise, solve the inequality $3{{\mathrm{e}}^{x}}&gt;x+3$.<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">Hence find $\displaystyle\displaystyle \displaystyle\int_{-2}^{2}{\left| 3{{\mathrm{e}}^{x}}-x-3 \right|\mathrm{d}x}$ , giving your answer in an exact form.<span class="inline-marks">[4]</span></div></li></ol>

Graphical Method

2010 JJC P1 Q10

Inequalities

<p>Solve the inequality $\frac{2}{4{{\left( x+1 \right)}^{2}}+1}>1$.</p><p style="text-align: right;">[2]</p>
<p>Hence find $\displaystyle\int_{-1}^{\frac{\sqrt{3}-2}{2}}{\left| 1-\frac{2}{4{{\left( x+1 \right)}^{2}}+1} \right|\,\mathrm{d}x}$, leaving your answer in exact form.</p><p style="text-align: right;">[3]</p>

Solve the inequality $\frac{2}{4{{\left( x+1 \right)}^{2}}+1}>1$.

Hence find $\displaystyle\displaystyle \displaystyle\int_{-1}^{\frac{\sqrt{3}-2}{2}}{\left| 1-\frac{2}{4{{\left( x+1 \right)}^{2}}+1} \right|\,\mathrm{d}x}$, leaving your answer in exact form.

Inequalities involving Rational Functions

Standard Integrals with Quadratic Denominators

2016 HCI P1 Q2

<ol class="i-list"><li><div class="list-item-paragraph">Given that $k$ is a positive constant, sketch the curve $E$ with equation $y=\left| {{x}^{2}}\left( x-k \right) \right|$, labelling the coordinates of the points where the curve cuts the axes.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Hence, find, in terms of $k$, the area of the region(s) bounded by $E$ and the $x$-axis for $0\le x\le 2k$.</div><p style="text-align: right;">[3]</p></li></ol>

Area Bounded by Curves

2024 NJC Promo Q3

<ol class="i-list"><li><div class="list-item-paragraph">Find ${\displaystyle \displaystyle\int xe^{2x} \mathrm{d}x}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence find the exact value of ${\displaystyle \displaystyle\int_{-1}^{1} \left|x\right|e^{2x} \mathrm{d}x}$.<span class="inline-marks">[3]</span></div></li></ol>

2024 MI PRU2 P1 Q3

<p>If $k$ is a non-zero integer, find $\displaystyle\int_{-\pi }^{\pi }{\sin kx\,\mathrm{d}x}$ and $\displaystyle\int_{-\pi }^{\pi }{\cos kx\,\mathrm{d}x}$.</p><p style="text-align: right;">[2]</p>
<p>Let $m$ and $n$ be positive integers.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that</div><div class="list-item-paragraph" style="text-align: center">$\displaystyle\int_{-\pi }^{\pi }{\sin \left( mx \right)\sin \left( nx \right)\mathrm{d}x=\left\{ \begin{aligned}
 \pi \mathrm{  if  }m&=n, \\
 0\mathrm{  if  }\,m&\ne n. \\
\end{aligned} \right.}$</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">By considering the formula for $\cos \left( mx+nx \right)$, evaluate</div><div class="list-item-paragraph" style="text-align: center">$\displaystyle\int_{-\pi }^{\pi }{\cos \left( mx \right)\cos \left( nx \right)\mathrm{d}x}$.</div></li></ol>
<p>Suppose $a$ is a positive integer.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Show that $\displaystyle\int_{-\pi }^{\pi }{\left| \sin \left( ax \right) \right|\mathrm{d}x}=4$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Hence, find the exact volume of the solid formed when the region bounded by the curve $y=\left| \sin \left( ax \right) \right|+1$, the $x$-axis and the lines $x=-\pi $ and $x=\pi $, is rotated completely about the $x$-axis.</div><p style="text-align: right;">[3]</p></li></ol>

If $k$ is a non-zero integer, find $\displaystyle\displaystyle \displaystyle\int_{-\pi }^{\pi }{\sin kx\,\mathrm{d}x}$ and $\displaystyle\displaystyle \displaystyle\int_{-\pi }^{\pi }{\cos kx\,\mathrm{d}x}$.

Volume of Revolution about the x-axis

Trigonometric Integration Using Product-to-Sum Formulae

2019 NJC P2 Q4

<ol class="abc-list"><li><div class="list-item-paragraph">Given that $n\ne 0$, show that $\displaystyle\int{x}\cos nx\,\mathrm{d}x=\frac{x\sin nx}{n}+\frac{\cos nx}{{{n}^{2}}}+c$, where $c$ is an arbitrary constant.</div></li><li><div class="list-item-paragraph">Using the result in part <strong>(a)</strong>, show that for all positive integers $n$, the value of $\displaystyle\int_{0}^{\pi }{x}\cos nx\,\mathrm{d}x$ can be expressed as $\frac{k}{{{n}^{2}}}$, where the possible value(s) of $k$ are to be determined.</div></li><li><div class="list-item-paragraph">Using the result in part <strong>(a)</strong> find the exact value of $\displaystyle\int_{0}^{\frac{\pi }{2}}{\left| x\cos 2x \right|}\,\mathrm{d}x$.</div></li></ol>

NJC Integration Techniques Tutorial Q9

<ol class="abc-list"><li><div class="list-item-paragraph">Find $\displaystyle\int{\frac{2x-1}{{{x}^{2}}+2x+1}\,}\mathrm{d}x$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find the exact value of $\displaystyle\int_{0}^{2}{\frac{\left| 2x-1 \right|}{{{x}^{2}}+2x+1}}\,\mathrm{d}x$.</div><p style="text-align: right;">[3]</p></li></ol>

Integration Using Partial Fractions

N2022 P1 Q8

<ol class="abc-list"><li data-list-item-id="item-1760430832089-08msq26t5" data-list-item-path="item-1760430832089-08msq26t5"><div class="list-item-paragraph">Find $\displaystyle\displaystyle \displaystyle\int{\frac{3-2x}{\sqrt{5+4x-{{x}^{2}}}}}\,\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760430832539-1dus2ppz9" data-list-item-path="item-1760430832539-1dus2ppz9"><div class="list-item-paragraph">Show that&nbsp;$\mathop{\displaystyle\displaystyle \displaystyle\int }_{\sqrt{2}}^{4}\left| x-3 \right|\mathrm{d}xx=a-b\sqrt{2}$, where&nbsp;$a$ and $b$ are constants to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760430833281-om5rnmmpz" data-list-item-path="item-1760430833281-om5rnmmpz"><div class="list-item-paragraph"><p>Using the substitution ${x=\sin u}$, find the exact value of ${\int_{\frac{\sqrt{2}}{2}}^{\frac{\sqrt{3}}{2}} \frac{1+x^2}{\sqrt{1-x^2}} \mathrm{d}x}$.</p><span class="inline-marks">[4]</span></div></li></ol>

Integration by Splitting the Numerator

Integration by Completing the Square

2023 VJC Promo Q6

<ol class="abc-list" start="2"><li><div class="list-item-paragraph">If $k>2$, find $\mathop{\displaystyle\int }_{0}^{k}\left| {{\mathrm{e}}^{x}}-2 \right|\,\mathrm{d}x$ in terms of $k$. Leave your answer in exact form.</div><p style="text-align: right;">[3]</p></li></ol>

2023 ASRJC Promo Q5

<p>For&nbsp;$a&gt;4$, show that&nbsp;$\displaystyle\displaystyle \displaystyle\int_{0}^{a}{\left| 2{{x}^{2}}-3x-2 \right|\,\,\mathrm{d}x}=\frac{1}{k}\left( 4{{a}^{3}}-9{{a}^{2}}-12a+56 \right)$ where&nbsp;$k$ is an integer to be found.<span class="inline-marks">[4]</span></p>

For $a>4$, show that $\displaystyle\displaystyle \displaystyle\int_{0}^{a}{\left| 2{{x}^{2}}-3x-2 \right|\,\,\mathrm{d}x}=\frac{1}{k}\left( 4{{a}^{3}}-9{{a}^{2}}-12a+56 \right)$ where $k$ is an integer to be found.

2024 DHS P1 Q1

<p>Without the use of a calculator, show that</p>
<p style="text-align: center">$\displaystyle\displaystyle \displaystyle\int_{-2}^{2}{\frac{\left| x-2 \right|}{{{x}^{2}}-2x+4}\mathrm{d}x=p\ln 3+q\pi }$,</p>
<p>where&nbsp;$p$ and $q$ are exact real constants to be determined.<span class="inline-marks">[6]</span></p>

Without the use of a calculator, show that

$\displaystyle\displaystyle \displaystyle\int_{-2}^{2}{\frac{\left| x-2 \right|}{{{x}^{2}}-2x+4}\mathrm{d}x=p\ln 3+q\pi }$,

where $p$ and $q$ are exact real constants to be determined.

2023 HCI P1 Q4

<ol class="abc-list"><li><div class="list-item-paragraph">The integral ${{I}_{n}}$ is defined by </div><div class="list-item-paragraph" style="text-align: center">\[
I_n = \displaystyle\int_{n\pi}^{(n+1)\pi} e^{-x}\,|\sin x|\,dx, for
\quad n \in \mathbb{N}.
\]</div><div class="list-item-paragraph">Show that</div><div class="list-item-paragraph" style="text-align: center">\[
I_0 = \frac{1}{2}\left( 1 + e^{-\pi} \right).
\]</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">By letting $y = x - n\pi$, show that</div><div class="list-item-paragraph" style="text-align: center">\[
I_n = e^{-n\pi} I_0.
\]</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence determine the exact value of $\displaystyle\int_{0}^{\infty }{{{\mathrm{e}}^{-x}}\left| \sin x \right|}\mathrm{ d}x$.</div><p style="text-align: right;">[5]</p></li></ol>

Improper Integrals and Convergence

Cyclic Integration by Parts

Exponential Integral Involving Absolute Sine Function

<p>Sketch the graph of $y=\frac{1}{{{\left( x-2 \right)}^{2}}}-1$ , showing clearly all asymptotes.<span class="inline-marks">[2]</span></p>
<p>The region $R$ is bounded by $x-$axis and the curve from $x=0$ to $x=\frac{3}{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the exact value of the area of $R$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The region $R$ is rotated $2\pi $ radians about the $y$-axis. Find the volume of the solid generated, giving your answer correct to $3$ decimal places.<span class="inline-marks">[4]</span></div></li></ol>

Sketch the graph of $y=\frac{1}{{{\left( x-2 \right)}^{2}}}-1$ , showing clearly all asymptotes.

The region $R$ is bounded by $x-$axis and the curve from $x=0$ to $x=\frac{3}{2}$.

Sketching Rational Functions

Volume of Revolution about the y-axis

Asymptotes and Discontinuities

2006 HCI Promo Q13

Graphing Techniques

<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find $\displaystyle\displaystyle \displaystyle\int{\frac{9}{\sqrt{3+2x-{{x}^{2}}}}}\mathrm{ d}x$. Hence, find the exact value of the constant $a$ for which $\displaystyle\displaystyle \displaystyle\int_{\ 0}^{\ 2}{\frac{9}{\sqrt{3+2x-{{x}^{2}}}}\mathrm{ d}x=\displaystyle\displaystyle \displaystyle\int_{\ -\tfrac{1}{2}}^{\ a}{\pi \left| \,1-2x\, \right|}}\mathrm{ d}x.$<span class="inline-marks">[7]</span></div></li></ol>

2013 RI P1 Q12

<ol class="abc-list"><li><div class="list-item-paragraph">Find $\displaystyle\displaystyle \displaystyle\int{\frac{3-2x}{\sqrt{8+2x-{{x}^{2}}}}}\ \mathrm{d}x$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Find the exact value of $\displaystyle\displaystyle \displaystyle\int_{-\frac{\pi }{8}}^{\frac{\pi }{6}}{\sec }2x\left| \tan 2x \right|\ \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the exact value of $\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi }{3}}{\frac{1}{1+8{{\cos }^{2}}x}}\ \mathrm{d}x$ using the substitution $t=\tan x$.<span class="inline-marks">[4]</span></div></li></ol>

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