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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

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

Find $\int \frac{1}{\mathrm{e}^x + 1} \mathrm{d}x$.

Integration

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

Find $\int \frac{\mathrm{e}^x}{\mathrm{e}^x + 1} \mathrm{d}x$.

<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-content"><span class="list-item-text">Express $\frac{1}{x^3 + x}$ as the sum of its partial fractions.</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-content"><span class="list-item-text">Hence find $\int \frac{1}{x^3 + x} \mathrm{d}x$</span></div></li></ol>

Express $\frac{1}{x^3 + x}$ as the sum of its partial fractions.

Hence find $\int \frac{1}{x^3 + x} \mathrm{d}x$

Integration Using Partial Fractions

Partial Fractions

Polynomials, Cubic Equations and Partial Fractions

<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>

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

<p>$\displaystyle \displaystyle\int{\frac{x+2}{{{x}^{2}}+4x+6}}\,\mathrm{d}x$</p>

$\displaystyle \displaystyle\int{\frac{x+2}{{{x}^{2}}+4x+6}}\,\mathrm{d}x$

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find $\displaystyle \displaystyle\int \frac{x}{\sqrt{x^2 - 4}} \, \mathrm{d}x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find $\displaystyle \displaystyle\int x^2 \ln (2x) \, \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find $\displaystyle \displaystyle\int \frac{\sin 2x}{1 + \cos^2 x} \, \mathrm{d}x$.<span class="inline-marks">[2]</span></div></li></ol>

Integration by Parts

2025 MI P1 Q3

Applications of Integration

<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

<p>Solve $\displaystyle\int{\frac{{e}^{2x}}{1-{{e}^{2x}}}}\mathrm{d}x$.</p>

Solve $\displaystyle\displaystyle \displaystyle\int{\frac{{e}^{2x}}{1-{{e}^{2x}}}}\mathrm{d}x$.

Worked Example 1

<ol class="i-list"><li><div class="list-item-paragraph">Find the integral $\displaystyle\int 3x^{2} \sqrt{x^{3} + 5}$ $dx$.</div></li><li><div class="list-item-paragraph">Find the integral $\displaystyle\int \frac{16x}{(4x^{2} + 4)^{2}}$ $dx$.</div></li><li><div class="list-item-paragraph">Evaluate the integral $\displaystyle\int \frac{2x - 3}{\sqrt{x^{2} - 3x + 1}}$ $dx$.</div></li><li><div class="list-item-paragraph">Evaluate the integral $\displaystyle\int -\frac{8x}{(2x^{2} + 3)^{2}}$ $dx$.</div></li><li><div class="list-item-paragraph">$\displaystyle\int \frac{x}{x^{2} + 1}$ $dx$</div></li><li><div class="list-item-paragraph">$\displaystyle\int \frac{2x^{3}}{3x^{4} + 2}$ $dx$</div></li><li><div class="list-item-paragraph">$\displaystyle\int \sin 5x (2 - \cos 5x)^{5}$ $dx$</div></li><li><div class="list-item-paragraph">$\displaystyle\int \frac{\sin 2x}{3 - \cos 2x}$ $dx$</div></li></ol>

u-Substitution Method (Cancelling Method)

<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 $\displaystyle \displaystyle\int{\frac{\mathrm{f}\left( x \right)}{\mathrm{g}\left( x \right)}\,\mathrm{d}x}$

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

<p>$\displaystyle\int{\frac{1}{x\ln x}}\mathrm{ d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{1}{x\ln x}}\mathrm{ d}x$

Reverse Chain Rule with Logarithmic and Radical Inner Functions

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

$\displaystyle\displaystyle \displaystyle\int{\frac{{{e}^{-3x}}}{3-{{e}^{-3x}}}}\mathrm{ d}x$

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

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

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

$\displaystyle\displaystyle \displaystyle\int{\frac{{{e}^{x}}}{{{e}^{-x}}+{{e}^{x}}}}\mathrm{ d}x$

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

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

Math Question Bank