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A Math (IGCSE)

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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

<ol class="abc-list list-columns-2" data-list-columns="2" data-list-flow="column"><li data-list-item-id="item-1787074918782-eg1jkwaa5" data-list-item-path="item-1787074918782-eg1jkwaa5" style="grid-column: 1; grid-row: 1;"><div class="list-item-paragraph"><p>$\displaystyle \displaystyle \displaystyle\int{{{x}^{2}}{{({{x}^{3}}+3)}^{3}}}\,\mathrm{d}x$</p></div></li><li data-list-item-id="item-1787074955422-xdz0eoruj" data-list-item-path="item-1787074955422-xdz0eoruj" style="grid-column: 1; grid-row: 2;"><div class="list-item-paragraph"><p>$\displaystyle \displaystyle \displaystyle\int{(5{{x}^{4}}+4x)({{x}^{5}}+2{{x}^{2}}}+1{{)}^{3}}\,\mathrm{d}x$</p></div></li><li data-list-item-id="item-1787074966558-y8nv8u0g3" data-list-item-path="item-1787074966558-y8nv8u0g3" style="grid-column: 1; grid-row: 3;"><div class="list-item-paragraph"><p>$\displaystyle \displaystyle \displaystyle\int{\frac{3x-2}{{{\left( 3{{x}^{2}}-4x-1 \right)}^{3}}}}\,\mathrm{d}x$</p></div></li><li data-list-item-id="item-1787075040359-5v8laelox" data-list-item-path="item-1787075040359-5v8laelox" style="grid-column: 1; grid-row: 4;"><div class="list-item-paragraph"><p>$\displaystyle \displaystyle \displaystyle\int{\frac{{{x}^{2}}}{\sqrt{{{x}^{3}}+1}}\,\mathrm{d}x}$ </p></div></li><li data-list-item-id="item-1787075051473-jvo81iutj" data-list-item-path="item-1787075051473-jvo81iutj" style="grid-column: 1; grid-row: 5;"><div class="list-item-paragraph"><p>$\displaystyle \displaystyle \displaystyle\int{\frac{21{{x}^{2}}+14}{\sqrt{{{x}^{3}}+2x+1}}\,\mathrm{d}x}$</p></div></li><li data-list-item-id="item-1787075064040-viw4ca6vv" data-list-item-path="item-1787075064040-viw4ca6vv" style="grid-column: 2; grid-row: 1;"><div class="list-item-paragraph"><p>$\displaystyle \displaystyle \displaystyle\int{\frac{x}{{{\sqrt{1-2{{x}^{2}}}}^{5}}}\,\mathrm{d}x}$</p></div></li><li data-list-item-id="item-1787075092632-q54lic0m0" data-list-item-path="item-1787075092632-q54lic0m0" style="grid-column: 2; grid-row: 2;"><div class="list-item-paragraph"><p>$\displaystyle \displaystyle \displaystyle\int{{{\left( \frac{x}{\sqrt{3{{x}^{4}}-1}} \right)}^{3}}\,\mathrm{d}x}$</p></div></li><li data-list-item-id="item-1787075106784-v2pxwt1ua" data-list-item-path="item-1787075106784-v2pxwt1ua" style="grid-column: 2; grid-row: 3;"><div class="list-item-paragraph"><p>$\displaystyle \displaystyle \displaystyle\int{3{{x}^{3}}{{(2{{x}^{4}}-5)}^{8}}}\,\mathrm{d}x$</p></div></li><li data-list-item-id="item-1787075114848-bpmjbbn6z" data-list-item-path="item-1787075114848-bpmjbbn6z" style="grid-column: 2; grid-row: 4;"><div class="list-item-paragraph"><p>$\displaystyle \displaystyle \displaystyle\int{{{e}^{3x}}{{({{e}^{3x}}+5)}^{11}}}\,\mathrm{d}x$</p></div></li><li data-list-item-id="item-1787075124273-jeym71jxc" data-list-item-path="item-1787075124273-jeym71jxc" style="grid-column: 2; grid-row: 5;"><div class="list-item-paragraph"><p>$\displaystyle \displaystyle \displaystyle\int{\frac{{{x}^{3}}}{{{x}^{4}}+3}\text{}\hspace{0.5em}}\mathrm{d}x$</p></div></li></ol>

Integration by the Reverse Chain Rule

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

Techniques of Integration

<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

Integrals Involving Long Division

Indefinite Integrals by Inspection

Integration by Completing the Square

Standard Integrals with Quadratic Denominators

<ol class="abc-list" start="2"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Find $\int x\left[ (1-3x^2)^5 + \mathrm{e}^{x^2+1} \right] \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph">Find $\frac{\mathrm{d}}{\mathrm{d}x}\cos^{-1}(x^2)$. Hence, or otherwise, find the exact value of $\int_{0}^{\frac{1}{\sqrt{2}}} \frac{x}{\sqrt{(1-x^4)}} \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Find $\int x\left[ (1-3x^2)^5 + \mathrm{e}^{x^2+1} \right] \mathrm{d}x$.

Find $\frac{\mathrm{d}}{\mathrm{d}x}\cos^{-1}(x^2)$. Hence, or otherwise, find the exact value of $\int_{0}^{\frac{1}{\sqrt{2}}} \frac{x}{\sqrt{(1-x^4)}} \mathrm{d}x$.

Derivatives of Inverse Trigonometric Functions

Integration as an Inverse Process of Differentiation

Chain Rule

2016 DHS C1BT Q2

Techniques of Differentiation

<p>Find $\int \frac{\sin(\ln(x))}{x} \mathrm{d}x$.</p>

Find $\int \frac{\sin(\ln(x))}{x} \mathrm{d}x$.

Integration

<p>Find $\int \mathrm{e}^{\cos(t)} \sin(t) \mathrm{d}t$.</p>

Find $\int \mathrm{e}^{\cos(t)} \sin(t) \mathrm{d}t$.

<p>Find $\int e^x \sin(e^x) \mathrm{d}x$.</p>

<p>Find $\int \frac{x}{(x^2 + 1)^2} \mathrm{d}x$.</p>

Find $\int \frac{x}{(x^2 + 1)^2} \mathrm{d}x$.

<p>Find $\int x^2 (x^3 + 5)^9 \mathrm{d}x$.</p>

Find $\int x^2 (x^3 + 5)^9 \mathrm{d}x$.

<p>Find $\int \frac{\sin \left(\frac{1}{x}\right)}{x^{2}} \mathrm{d} x$.</p>

Find $\int \frac{\sin \left(\frac{1}{x}\right)}{x^{2}} \mathrm{d} x$.

<p>Find $\int \sec^2 x \sqrt{\tan x + 1} \, \mathrm{d}x$ .</p>

Find $\int \sec^2 x \sqrt{\tan x + 1} \, \mathrm{d}x$ .

<p>Find $\int e^{3x} \sqrt{e^{3x} + 2} \mathrm{~d}x$.</p>
<p style="text-align: right"></p>

Find $\int e^{3x} \sqrt{e^{3x} + 2} \mathrm{~d}x$.

<p>Find $\int x \sin(2x^2 + 3) \mathrm{d}x$.</p>

Find $\int x \sin(2x^2 + 3) \mathrm{d}x$.

<p>$\displaystyle\int{{{e}^{2x}}\sqrt{{{e}^{2x}}+1}}\,\mathrm{d}x$</p>

$\displaystyle\int{{{e}^{2x}}\sqrt{{{e}^{2x}}+1}}\,\mathrm{d}x$

<p>$\displaystyle\int{x\sin \left( 3{{x}^{2}}+1 \right)}\,\mathrm{d}x$</p>

$\displaystyle\int{x\sin \left( 3{{x}^{2}}+1 \right)}\,\mathrm{d}x$

<p>$\displaystyle \displaystyle\int{{{x}^{2}}\sin (3{{x}^{3}})}\,\mathrm{d}x$</p>

$\displaystyle \displaystyle\int{{{x}^{2}}\sin (3{{x}^{3}})}\,\mathrm{d}x$

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

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

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

$\displaystyle \displaystyle\int{x{{e}^{{{x}^{2}}+1}}}\,\mathrm{d}x$

<p>$\displaystyle \displaystyle\int{4{{x}^{3}}}{{({{x}^{4}}+7)}^{4}}\,\mathrm{d}x$</p>

$\displaystyle \displaystyle\int{4{{x}^{3}}}{{({{x}^{4}}+7)}^{4}}\,\mathrm{d}x$

<ol class="abc-list"><li data-list-item-id="item-a-4" data-list-item-path="item-a-4"><div class="list-item-content"><span class="list-item-text">Use the substitution $x=\sin^2 \theta$, where&nbsp;$0 \leq \theta \leq \frac{\pi}{2}$, to find $\displaystyle \displaystyle\int_0^{\frac{1}{2}} \sqrt{\frac{16x}{1-x}} \, \mathrm{d}x$ exactly.</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b-4" data-list-item-path="item-b-4"><div class="list-item-content"><span class="list-item-text">Find $\displaystyle \displaystyle\int_a^b \frac{|x-1|}{\sqrt{\frac{1}{2}x^2 - x + 1}} \, \mathrm{d}x$ in terms of $a$ and $b$, where&nbsp;$a &lt; 1 &lt; b$.</span><span class="inline-marks">[4]</span></div></li></ol>

Use the substitution $x=\sin^2 \theta$, where $0 \leq \theta \leq \frac{\pi}{2}$, to find $\displaystyle \displaystyle\int_0^{\frac{1}{2}} \sqrt{\frac{16x}{1-x}} \, \mathrm{d}x$ exactly.

Find $\displaystyle \displaystyle\int_a^b \frac{|x-1|}{\sqrt{\frac{1}{2}x^2 - x + 1}} \, \mathrm{d}x$ in terms of $a$ and $b$, where $a < 1 < b$.

Math Question Bank