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<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

<p>$\displaystyle\int{\frac{1}{\sqrt{7+10x-{{x}^{2}}}}}\mathrm{ d}x$</p>

$\displaystyle\int{\frac{1}{\sqrt{7+10x-{{x}^{2}}}}}\mathrm{ d}x$

<p>Without the use of a calculator, show that</p>
<p style="text-align: center">\[\int_{2-\sqrt{3}}^{5}\frac{\left|x-5\right|}{x^2-4x+7}\,\mathrm{d}x=p\ln 2+q\pi,\]</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

\[\int_{2-\sqrt{3}}^{5}\frac{\left|x-5\right|}{x^2-4x+7}\,\mathrm{d}x=p\ln 2+q\pi,\]

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

Exact Definite Integral with a Modulus

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

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

<p>$\displaystyle \displaystyle\int{\frac{3x+7}{-3{{x}^{2}}+4x+6}}\,\mathrm{d}x$</p>

$\displaystyle \displaystyle\int{\frac{3x+7}{-3{{x}^{2}}+4x+6}}\,\mathrm{d}x$

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

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

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

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

<p>$\int_{-2}^{\alpha-1} \frac{1}{\left(1-2x-x^2\right)^{\frac{3}{2}}}\,\mathrm{d}x\quad \left(x+1=\sqrt{2}\sin\theta,\ 0 &lt; \alpha &lt; 1\right)$</p>

$\int_{-2}^{\alpha-1} \frac{1}{\left(1-2x-x^2\right)^{\frac{3}{2}}}\,\mathrm{d}x\quad \left(x+1=\sqrt{2}\sin\theta,\ 0 < \alpha < 1\right)$

Integration by Substitution

HCI Integration Tutorial Q4 (g)

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

$\displaystyle\int{\frac{1}{\sqrt{2+8x-{{x}^{2}}}}}\mathrm{ d}x$

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

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

<ol class="abc-list"><li data-list-item-id="item-a-3" data-list-item-path="item-a-3"><ol class="i-list"><li data-list-item-id="item-a-3-i" data-list-item-path="item-a-3/item-a-3-i"><div class="list-item-paragraph"><p>Show that&nbsp;$\frac{3x^2}{(x+1)(3x^2+x+1)}$ can be expressed as<br></p></div><div class="list-item-paragraph"><p style="text-align: center;">$\frac{A}{x+1} + \frac{Bx+C}{3x^2+x+1}$,<br></p></div><div class="list-item-paragraph"><p>where&nbsp;$A$, $B$ and $C$ are constants to be determined.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-3-ii" data-list-item-path="item-a-3/item-a-3-ii"><div class="list-item-paragraph">Hence, find $\displaystyle \displaystyle\int \frac{3x^2}{(x+1)(3x^2+x+1)} \, \mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>
<ol class="abc-list" start="2"><li data-list-item-id="item-b-3" data-list-item-path="item-b-3"><div class="list-item-paragraph">The region $R$ is bounded by the curve $y = \frac{3x^2}{(x+1)(3x^2+x+1)}$, the line $x=4$ and the $x$-axis.</div><ol class="i-list"><li data-list-item-id="item-b-3-i" data-list-item-path="item-b-3/item-b-3-i"><div class="list-item-paragraph">Sketch the graph of $y = \frac{3x^2}{(x+1)(3x^2+x+1)}$ for&nbsp;$x \ge 0$, giving the coordinates of any intercepts with the axes and the equations of any asymptotes. Shade the region $R$ on your sketch.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-3-ii" data-list-item-path="item-b-3/item-b-3-ii"><div class="list-item-paragraph">Find the exact area of $R$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-3-iii" data-list-item-path="item-b-3/item-b-3-iii"><div class="list-item-paragraph">Find the volume of the solid generated when&nbsp;$R$ is rotated through $2\pi$ radians about the $x$-axis.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Sketching Rational Functions

Integration Using Partial Fractions

Volume of Revolution about the x-axis

Area Bounded by Curves

Asymptotes and Discontinuities

Partial Fractions

2025 CJC P1 Q11

Applications of Integration

Graphing Techniques

<ol class="abc-list"><li><div class="list-item-paragraph">Solve the following integral.</div><ol class="i-list"><li><div class="list-item-paragraph">$\displaystyle \displaystyle\int \frac{1-2x}{\sqrt{3+2x-x^2}} \, \mathrm{d}x$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">$\displaystyle \displaystyle\int x \ln(2-x^2) \, \mathrm{d}x$, where&nbsp;$-\sqrt{2} &lt; x &lt; \ \sqrt{2}$.<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph">A function $\mathrm{f}$ is defined by $\mathrm{f}(x) = \mathrm{e}^x - 2$.</div><ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y = |\mathrm{f}(x)|$, indicating clearly the equation(s) of asymptote(s), if any.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence find $\displaystyle \displaystyle\int_0^{\ln 3} |\mathrm{f}(x)| \, \mathrm{d}x$, giving your answer in exact form.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Sketching Exponential Functions and Solving Equations Graphically

Integration by Parts

Modulus Functions

Graphs of Modulus Function

2025 ASRJC P2 Q5

Functions

Transformations of Graphs

<p>Solve $\displaystyle\int{\frac{1}{\sqrt{1+6x-{{x}^{2}}}}}\mathrm{d}x$.</p>

Solve $\displaystyle\displaystyle \displaystyle\int{\frac{1}{\sqrt{1+6x-{{x}^{2}}}}}\mathrm{d}x$.

Worked Example 3

<ol class="i-list"><li><div class="list-item-paragraph">$\displaystyle\int{\frac{5}{{{x}^{2}}+5}}\mathrm{ d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int{\frac{1}{{{x}^{2}}+7}}\mathrm{ d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int{\frac{11}{{{x}^{2}}-4x+10}}\mathrm{ d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int{\frac{3}{{{x}^{2}}+5x+5}}\mathrm{ d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int{\frac{3}{\sqrt{8x-{{x}^{2}}+1}}}\mathrm{ d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int{\frac{4}{\sqrt{5-4x-{{x}^{2}}}}}\mathrm{ d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int{\frac{1}{4{{x}^{2}}-x-3}}\mathrm{ d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int{\frac{2}{3{{x}^{2}}+14x-5}}\mathrm{ d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int{\frac{1}{5{{x}^{2}}-x-5}}\mathrm{ d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int{\frac{1}{3{{x}^{2}}-2x-4}}\mathrm{ d}x$</div></li></ol>

Using Formulae from MF 27

<p>$\displaystyle\int{\frac{5-3x}{\sqrt{7-5x-{{x}^{2}}}}\mathrm{ d}x}$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{5-3x}{\sqrt{7-5x-{{x}^{2}}}}\mathrm{ d}x}$

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

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

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

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

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

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

<p>$\displaystyle\int{\frac{1}{\sqrt{1+6x-{{x}^{2}}}}}\mathrm{ d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{1}{\sqrt{1+6x-{{x}^{2}}}}}\mathrm{ d}x$

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