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E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li data-list-item-id="item-1785847295068-a8zsodu8n" data-list-item-path="item-1785847295068-a8zsodu8n"><div class="list-item-paragraph"><p>$\int \frac{1}{4-9x} \mathrm{d}x$</p></div></li><li data-list-item-id="item-1785847297899-q5i719d1n" data-list-item-path="item-1785847297899-q5i719d1n"><div class="list-item-paragraph"><p>$\int \frac{1}{4-9x^2} \mathrm{d}x$</p></div></li><li data-list-item-id="item-1785847299155-yqtbf39aq" data-list-item-path="item-1785847299155-yqtbf39aq"><div class="list-item-paragraph"><p>$\int \frac{x}{4-9x^2} \mathrm{d}x$</p></div></li><li data-list-item-id="item-1785847299892-ahmcmu9lj" data-list-item-path="item-1785847299892-ahmcmu9lj"><div class="list-item-paragraph"><p>$\int \frac{x+3}{4-9x^2} \mathrm{d}x$</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="5"><li data-list-item-id="item-1785847311604-x9nrhgskw" data-list-item-path="item-1785847311604-x9nrhgskw"><div class="list-item-paragraph"><p>$\int \frac{x^2}{4-9x^2} \mathrm{d}x$</p></div></li><li data-list-item-id="item-1785847312891-wh8uh9njz" data-list-item-path="item-1785847312891-wh8uh9njz"><div class="list-item-paragraph"><p>$\int \frac{x}{\sqrt{4-9x^2}} \mathrm{d}x$</p></div></li><li data-list-item-id="item-1785847314123-msl6ulyl3" data-list-item-path="item-1785847314123-msl6ulyl3"><div class="list-item-paragraph"><p>Find $\int \frac{x+2}{\sqrt{1-8x-4x^2}} \mathrm{d}x$.</p></div></li><li data-list-item-id="item-1785847314843-jeq1sfc48" data-list-item-path="item-1785847314843-jeq1sfc48"><div class="list-item-paragraph"><p>Find $\int \frac{x}{x^2+4x+7} \mathrm{d}x$.</p></div></li></ol></div></div></div>

Integration by Splitting the Numerator

Integration by the Reverse Chain Rule

Integrals Involving Long Division

Indefinite Integrals by Inspection

Logarithmic Integrals of $\frac{f'(x)}{f(x)}$

Integration by Completing the Square

Standard Integrals with Quadratic Denominators

Techniques of Integration

<ol class="i-list"><li data-list-item-id="amint-q03-question-1" data-list-item-path="amint-q03-question-1"><div class="list-item-paragraph">Express $\dfrac{2x^3-3x^2-3x-6}{x(x-2)}$ in the form $ax+b+\dfrac{c}{x}+\dfrac{d}{x-2}$, where&nbsp;$a$, $b$, $c$ and $d$ are integers.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q03-question-2" data-list-item-path="amint-q03-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of $\displaystyle\int_3^5 \dfrac{2x^3-3x^2-3x-6}{x(x-2)}\,\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol>

Express $\dfrac{2x^3-3x^2-3x-6}{x(x-2)}$ in the form $ax+b+\dfrac{c}{x}+\dfrac{d}{x-2}$, where $a$, $b$, $c$ and $d$ are integers.

Hence evaluate the exact value of $\displaystyle\int_3^5 \dfrac{2x^3-3x^2-3x-6}{x(x-2)}\,\mathrm{d}x$.

Integration Using Partial Fractions

Partial Fractions

Polynomial Division and Remainders

Definite Integral

Improper rational definite integral

Integration

Polynomials, Cubic Equations and Partial Fractions

<p>$\displaystyle \displaystyle\int{\frac{4{{x}^{2}}+2}{2{{x}^{2}}-x-1}}\,\mathrm{d}x$</p>

$\displaystyle \displaystyle\int{\frac{4{{x}^{2}}+2}{2{{x}^{2}}-x-1}}\,\mathrm{d}x$

<p>$\displaystyle \displaystyle\int{\frac{{{x}^{2}}-2x+6}{{{x}^{2}}+2x+6}}\,\mathrm{d}x$</p>

$\displaystyle \displaystyle\int{\frac{{{x}^{2}}-2x+6}{{{x}^{2}}+2x+6}}\,\mathrm{d}x$

<p>$\displaystyle \displaystyle\int{\frac{5{{x}^{4}}-5{{x}^{2}}+3x}{1-{{x}^{2}}}}\,\mathrm{d}x$</p>

$\displaystyle \displaystyle\int{\frac{5{{x}^{4}}-5{{x}^{2}}+3x}{1-{{x}^{2}}}}\,\mathrm{d}x$

<p>$\displaystyle \displaystyle\int{\frac{12-40x+40{{x}^{2}}-8{{x}^{3}}}{2{{x}^{2}}-8x+1}}\,\mathrm{d}x$</p>

$\displaystyle \displaystyle\int{\frac{12-40x+40{{x}^{2}}-8{{x}^{3}}}{2{{x}^{2}}-8x+1}}\,\mathrm{d}x$

<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 $\displaystyle\displaystyle \displaystyle\int \frac{1}{\sqrt{x} + \sqrt[3]{x}} \, \mathrm{d}x$.

Integrals involving Modulus Functions

Integrating Powers of Sine and Cosine

Integrable Forms of Trigonometric Functions

Integration by Substitution

2025 EJC P1 Q10

Applications of Integration

<p>$\displaystyle\int \frac{e^{-2x}}{e^{-x} + 1} dx$, let&nbsp;$u = e^{-x}$.</p>

$\displaystyle\displaystyle \displaystyle\int \frac{e^{-2x}}{e^{-x} + 1} dx$, let $u = e^{-x}$.

<p>$\displaystyle\displaystyle \displaystyle\int{\frac{{{x}^{2}}+2}{{{x}^{2}}+2x+3}\mathrm{ }}\mathrm{d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{{{x}^{2}}+2}{{{x}^{2}}+2x+3}\mathrm{ }}\mathrm{d}x$

<p>$\displaystyle\displaystyle \displaystyle\int{\frac{3{{x}^{2}}+9x+5}{3{{x}^{2}}+5x+2}\mathrm{ }}\mathrm{d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{3{{x}^{2}}+9x+5}{3{{x}^{2}}+5x+2}\mathrm{ }}\mathrm{d}x$

<p>$\displaystyle\int{\frac{{{x}^{4}}+4{{x}^{2}}+1}{4+{{x}^{2}}}}\mathrm{ d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{{{x}^{4}}+4{{x}^{2}}+1}{4+{{x}^{2}}}}\mathrm{ d}x$

<p>$\displaystyle\int{\frac{{{x}^{2}}+5x+4}{{{x}^{2}}+4}}dx$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{{{x}^{2}}+5x+4}{{{x}^{2}}+4}}dx$

<p class="paragraph-header">Always do long division first when the degree of $\mathrm{f}(x)\ge $ degree of $\mathrm{g}(x)$ for&nbsp;$\displaystyle\int{\frac{\mathrm{f}\left( x \right)}{\mathrm{g}\left( x \right)}\,\mathrm{d}x}$</p>
<p>$\displaystyle\int{\frac{{{x}^{4}}-{{x}^{2}}+x}{{{x}^{3}}+1}}dx$</p>

Always do long division first when the degree of $\mathrm{f}(x)\ge $ degree of $\mathrm{g}(x)$ for $\displaystyle \displaystyle\int{\frac{\mathrm{f}\left( x \right)}{\mathrm{g}\left( x \right)}\,\mathrm{d}x}$

$\displaystyle\displaystyle \displaystyle\int{\frac{{{x}^{4}}-{{x}^{2}}+x}{{{x}^{3}}+1}}dx$

<ol class="abc-list"><li data-list-item-id="item-1760955635047-9fe3nmrk6" data-list-item-path="item-1760955635047-9fe3nmrk6"><div class="list-item-paragraph">Using the substitution $u=\sqrt{4x+1}$, show that$\displaystyle\displaystyle \displaystyle\int{\frac{\sqrt{4x+1}}{x-2}}\mathrm{ d}x=\displaystyle\displaystyle \displaystyle\int{\frac{2{{u}^{2}}}{{{u}^{2}}-9}}\mathrm{ d}u$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760955635537-h5mrbedmj" data-list-item-path="item-1760955635537-h5mrbedmj"><div class="list-item-paragraph">The region $R$ lies in the first quadrant and is bounded by the curve $y=\frac{\sqrt{4x+1}}{x-2}$ and the lines $x=6$, $x=12$ and $y=0.5$.</div><ol class="i-list"><li data-list-item-id="subitem-1760955652450-22oklxjp5" data-list-item-path="item-1760955635537-h5mrbedmj/subitem-1760955652450-22oklxjp5"><div class="list-item-paragraph"><p>Sketch the graph of $y=\frac{\sqrt{4x+1}}{x-2}$ for&nbsp;$x \geqslant 0$, giving the coordinates of any intercepts with the axes and the equations of any asymptotes. Shade the region $R$ on your sketch.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760955653573-0vzpnih3e" data-list-item-path="item-1760955635537-h5mrbedmj/subitem-1760955653573-0vzpnih3e"><div class="list-item-paragraph">Find the exact area of $R$, giving your answer in the form $a+b\ln c$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="subitem-1760955654465-kpwamcczy" data-list-item-path="item-1760955635537-h5mrbedmj/subitem-1760955654465-kpwamcczy"><div class="list-item-paragraph"><p>Find the volume of the solid generated when&nbsp;$R$ is rotated through $2\pi $ radians about the $x$- axis.</p><p>Give your answer correct to $2$ decimal places.</p><span class="inline-marks">[3]</span></div></li></ol></li></ol>

Volume of Revolution about the x-axis

Asymptotes and Discontinuities

Area bounded by 2 curves

N2024 P1 Q10

Graphing Techniques

<p>The region bounded by the curve with equation $x=\frac{y}{\sqrt{2y-{{y}^{2}}}}$, the lines $y=1$, $y=1.6$ and the $y$-axis is rotated through $2\pi $ radians about the $y$-axis to form a solid ornament.</p>
<ol class="abc-list"><li data-list-item-id="item-1760765492274-msblxptdn" data-list-item-path="item-1760765492274-msblxptdn"><div class="list-item-paragraph">Find the exact volume ${{V}_{1}}$ of the ornament, giving your answer in terms of $\pi $.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760765492769-m3fqs86cx" data-list-item-path="item-1760765492769-m3fqs86cx"><div class="list-item-paragraph">An ornament designer designs a different ornament by rotating the region bounded by another curve with equation $x=\frac{by}{\sqrt{2y-{{y}^{2}}}}$, where&nbsp;$b&gt;0$, the lines $y=1$, $y=1.6$ and the $y$-axis. The region is rotated through $2\pi $ radians about the $y$-axis. The volume generated is now ${{V}_{2}}$. State the ratio of ${{V}_{1}}$ to ${{V}_{2}}$.<span class="inline-marks">[1]</span></div></li></ol>

The region bounded by the curve with equation $x=\frac{y}{\sqrt{2y-{{y}^{2}}}}$, the lines $y=1$, $y=1.6$ and the $y$-axis is rotated through $2\pi $ radians about the $y$-axis to form a solid ornament.

Volume of Revolution about the y-axis

2024 HCI P1 Q3

<p>A curve ${C}$ has equation ${y = \frac{3 - x}{x - 1}}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Sketch ${C}$, stating the equations of the asymptotes.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the exact volume of the solid generated when the region bounded by ${C}$, the ${x}$-axis and the line ${x=9}$ is rotated through ${2\pi}$ radians about the ${x}$-axis. Leave your answer in the form ${\pi (a + b \ln 2)}$, where ${a}$ and ${b}$ are constants to be determined.</div><p style="text-align: right;">[5]</p></li></ol>

A curve ${C}$ has equation ${y = \frac{3 - x}{x - 1}}$.

Sketching Rational Functions

2024 TJC P2 Q1

<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)**.

Tangents and Normals to Parametric Curves

Quotient Rule

Differentiation of Parametric Equations

Chain Rule

Converting Parametric Equations to Cartesian Form

2019 NYJC P2 Q2

Techniques of Differentiation

Applications of Differentiation

<ol class="abc-list"><li data-list-item-id="item-1760705839166-6wgslxx7s"><ol class="i-list"><li data-list-item-id="subitem-1760705844948-unqmkvr51"><div class="list-item-paragraph">Use the substitution $u=1+{{x}^{2}}$ to find $\displaystyle\displaystyle \displaystyle\int{\frac{x{{\mathrm{e}}^{1+{{x}^{2}}}}}{\sqrt{1+{{\mathrm{e}}^{1+{{x}^{2}}}}}}\,}\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="subitem-1760705850528-s8yo46c3h"><div class="list-item-paragraph">Curves ${{C}_{1}}$ and ${{C}_{2}}$ have equations $y=x{{\mathrm{e}}^{{{x}^{2}}-2}}-\frac{1}{2\mathrm{e}}$ and $y=\frac{x{{\mathrm{e}}^{1+{{x}^{2}}}}}{\sqrt{1+{{\mathrm{e}}^{1+{{x}^{2}}}}}}-\frac{1}{2{{\mathrm{e}}^{2}}}$ respectively. The region bounded by the curves ${{C}_{1}}$ and ${{C}_{2}}$, the $y$-axis and the line $x=1$ is $R$.<br>Find the exact area of $R$.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1760705839796-w6veplpft"><div class="list-item-paragraph">The shape of a vase is formed by rotating the part of the curve $x=\sqrt{\frac{2{{y}^{3}}-y+1}{2{{y}^{2}}+1}}$ between $y=0$ and $y=1$ through $2\pi $ radians about the $y$-axis (see diagram below). Find the exact volume of the vase formed.<span class="inline-marks">[5]</span></div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760705881411-mieva30jfy.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA3MDU4ODE0MTEtbWlldmEzMGpmeS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODI2Mjc1LCJleHAiOjE3OTI4MjY1NzV9.-83g-CLv_l3IZ3p0ymQqerLCgtI9TuiKxwAL-lBw7t4" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div></li></ol>

2022 ASRJC P2 Q3

<ol class="abc-list"><li data-list-item-id="item-1760427949744-c0hfli8jo" data-list-item-path="item-1760427949744-c0hfli8jo"><div class="list-item-paragraph">Find $\displaystyle\displaystyle \displaystyle\int{\frac{\cos x}{\sqrt[3]{\sin x}}}\,\mathrm{d}x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760427950546-9shtacm0d" data-list-item-path="item-1760427950546-9shtacm0d"><div class="list-item-paragraph">Using the substitution $x=1-{{\mathrm{e}}^{u}}$, find $\displaystyle\displaystyle \displaystyle\int{\frac{1}{1-{{\mathrm{e}}^{u}}}\,\mathrm{d}u}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760427951094-nv5czjhex" data-list-item-path="item-1760427951094-nv5czjhex"><div class="list-item-paragraph">Find $\displaystyle\displaystyle \displaystyle\int{x\ln \left( x+1 \right)\,\mathrm{d}x}$, where&nbsp;$x&gt;-1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760427952391-88zga1e48" data-list-item-path="item-1760427952391-88zga1e48"><ol class="i-list"><li data-list-item-id="subitem-1760427974629-fbyh0dpda" data-list-item-path="item-1760427952391-88zga1e48/subitem-1760427974629-fbyh0dpda"><div class="list-item-paragraph">Write down constants $A$ and $B$ such that, for all values of $x$, $x=A\left( 2x+2 \right)+B$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760427975567-fd4myxbnd" data-list-item-path="item-1760427952391-88zga1e48/subitem-1760427975567-fd4myxbnd"><div class="list-item-paragraph">Hence, find $\displaystyle\displaystyle \displaystyle\int{\frac{x}{{{x}^{2}}+2x+2}\,\mathrm{d}x}$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

Integration by Parts

2023 JPJC Promo Q13

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Find $\displaystyle\int{\frac{2-x}{\sqrt{(4x-{{x}^{2}}-1)}}\,\mathrm{d}x}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find $\displaystyle\int{\frac{1}{\sqrt{(4x-{{x}^{2}}-1)}}\,\mathrm{d}x}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Using the results of part <strong>(i)</strong> and <strong>(ii)</strong>, find $\displaystyle\int{\frac{x}{\sqrt{(4x-{{x}^{2}}-1)}}\,\mathrm{d}x}$.</div><p style="text-align: right;">[2]</p></li></ol></li><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">By using the substitution $u=\sqrt{x}$, find $\displaystyle\int{\frac{\sqrt{x}}{1+\sqrt{x}}\,\mathrm{d}x}$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find the exact value of $\displaystyle\int_{0}^{9}{\ln \left( 1+\sqrt{x} \right)\,\mathrm{d}x}$, leaving your answer in the form $a\ln b-\frac{c}{b}$, where $a$, $b$ and $c$ are integers to be determined.</div><p style="text-align: right;">[4]</p></li></ol></li></ol>

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