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

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

Integration by Parts

Integration by Substitution

Techniques of Integration

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">State the derivative of $\mathrm{e}^{\tan x}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">Hence, find $\int \mathrm{e}^{\tan x} (\sec^3 x)(\sin x) \,\mathrm{d}x$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Write down constants $A$ and $B$ such that, for all values of $x$, $1 - 4x = A(3 - 8x) + B$. Hence, find </div><div class="list-item-paragraph"><p>$\int \frac{1 - 4x}{\sqrt{1 - 4x^2 + 3x}} \,\mathrm{d}x.$</p><span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Use the substitution $u = \sqrt{1 + x}$ to find the exact value of $\int_{1}^{3} \frac{x^2}{\left(\sqrt{1 + x}\right)^3} \,\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol>

Derivatives of Trigonometric Functions

Integration as an Inverse Process of Differentiation

Derivatives of Exponential Functions

Chain Rule

2024 VJC Promo Q10

Techniques of Differentiation

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

2024 MI PRU2 P1 Q3

Applications of Integration

<p>In this question, $a$ is a positive constant.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Explain, with the aid of a sketch, why $\frac{a}{n} \sum_{k=1}^{n} \ln \left(1 + \frac{ak}{n}\right) &gt; \int_{0}^{a} \ln(1+x) \, \mathrm{d}x$ for all positive integers $n$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">State a similar expression that is less than $\int_{0}^{a} \ln(1+x) \, \mathrm{d}x$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Use your sketch in part <strong>(i)</strong> to explain why $\lim_{n \to \infty} \left[ \frac{a}{n} \sum_{k=1}^{n} \ln \left(1 + \frac{ak}{n}\right) \right] = \int_{0}^{a} \ln(1+x) \, \mathrm{d}x$ and find the value of this limit exactly in terms of $a$.<span class="inline-marks">[5]</span></div></li></ol>

In this question, $a$ is a positive constant.

Limit of an expression

Riemann Sums

Sigma Notation

Sequence and Series

<p>State the derivative of $\tan x$. Hence, find $\displaystyle\int \sec^3 x \, \mathrm{d}x$.<span class="inline-marks">[5]</span></p>

State the derivative of $\tan x$. Hence, find $\displaystyle\displaystyle\int \sec^3 x \, \mathrm{d}x$.

Integrable Forms of Trigonometric Functions

Cyclic Integration by Parts

2025 CJC CA1 Q3

<ol class="abc-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><ol class="i-list"><li data-list-item-id="imported-item-block-1-item-0-sub-0" data-list-item-path="imported-item-block-1-item-0/imported-item-block-1-item-0-sub-0"><div class="list-item-paragraph">Find $\int x^2 \cos n x \,\mathrm{d}x$, where&nbsp;$n$ is a positive integer.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-1-item-0-sub-1" data-list-item-path="imported-item-block-1-item-0/imported-item-block-1-item-0-sub-1"><div class="list-item-paragraph">Hence find $\int_{\pi}^{2\pi} x^2 \cos n x \,\mathrm{d}x$, giving your answers in the form $a\frac{\pi}{n^2}$, where the possible values of $a$ are to be determined.<span class="inline-marks">[2]</span></div></li></ol></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">The region bounded by the curve $y = \frac{x\sqrt{x}}{9-x^2}$, the $x$-axis and the lines $x = 0$ and $x = 2$ is rotated through $2\pi$ radians about the $x$-axis. Use the substitution $u = 9 - x^2$ to find the exact volume of the solid obtained, simplifying your answer.<span class="inline-marks">[5]</span></div></li></ol>

The region bounded by the curve $y = \frac{x\sqrt{x}}{9-x^2}$, the $x$-axis and the lines $x = 0$ and $x = 2$ is rotated through $2\pi$ radians about the $x$-axis. Use the substitution $u = 9 - x^2$ to find the exact volume of the solid obtained, simplifying your answer.

Volume of Revolution about the x-axis

N2016 P2 Q2 

<ol class="abc-list"><li data-list-item-id="item-1770367385574-v6t4x6m44" data-list-item-path="item-1770367385574-v6t4x6m44"><div class="list-item-paragraph">Evaluate $\displaystyle\int_{0}^{\infty }{x{{e}^{-4x}}\,\mathrm{d}x}$.</div></li><li data-list-item-id="item-1770367386444-y7jy9glkx" data-list-item-path="item-1770367386444-y7jy9glkx"><div class="list-item-paragraph">Hence, find $\displaystyle\int_{0}^{\infty }{{{x}^{2}}{{e}^{-4x}}\,\mathrm{d}x}$.</div></li></ol>

Improper Integrals and Convergence

<p>Express $\displaystyle\int_{1}^{\infty }{x{{e}^{-2x}}\,\mathrm{d}x} $ in the form of $a+b{{e}^{-2}}$ where&nbsp;$a$ and $b$ are constants to be determined.</p>

Express $\displaystyle\int_{1}^{\infty }{x{{e}^{-2x}}\,\mathrm{d}x} $ in the form of $a+b{{e}^{-2}}$ where $a$ and $b$ are constants to be determined.

<ol class="abc-list"><li data-list-item-id="item-a-2"><div class="list-item-content"><span class="list-item-text">Find $\displaystyle \displaystyle\int \frac{x}{\sqrt{16-x^4}} \, \mathrm{d}x$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-2"><div class="list-item-content"><span class="list-item-text">Find $\displaystyle \displaystyle\int \frac{x}{2} \sin^{-1} \left( \frac{x^2}{4} \right) \, \mathrm{d}x$.</span><span class="inline-marks">[5]</span></div></li></ol>

Find $\displaystyle \displaystyle\int \frac{x}{\sqrt{16-x^4}} \, \mathrm{d}x$.

Find $\displaystyle \displaystyle\int \frac{x}{2} \sin^{-1} \left( \frac{x^2}{4} \right) \, \mathrm{d}x$.

Reverse Chain Rule with Standard Integral Formulae

2025 RVHS P1 Q5

<ol class="abc-list"><li><div class="list-item-paragraph">Given that&nbsp;$\mathrm{f}$ is a continuous function, explain, with the aid of a sketch, why the value of\[\lim_{n \to \infty} \frac{2}{n} \left\{ \mathrm{f}\left(1 + \frac{2}{n}\right) + \mathrm{f}\left(1 + \frac{4}{n}\right) + \dots + \mathrm{f}\left(1 + \frac{2n}{n}\right) \right\}\]is $\displaystyle \displaystyle\int_1^3 \mathrm{f}(x) \, \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence evaluate $\lim\limits_{n \to \infty} \sum_{k=1}^n \ln\left(1 + \frac{2k}{n}\right) \frac{2}{n}$ exactly.<span class="inline-marks">[4]</span></div></li></ol>

2025 NYJC P2 Q3

Functions

<ol class="abc-list"><li data-list-item-id="item-1781605617444-65w2reoy4"><div class="list-item-paragraph">Find $\displaystyle \displaystyle\int \cos (3 \ln x) \, \mathrm{d}x$.<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">Let&nbsp;$I$ be the indefinite integral $\displaystyle \displaystyle\int \frac{\mathrm{P}(x)}{1-\sqrt{x}} \, \mathrm{d}x$, $0 &lt; x &lt; 1$, where&nbsp;$\mathrm{P}(x)$ is a polynomial in $x$.</span></div><ol class="i-list"><li data-list-item-id="item-b-i"><div class="list-item-content"><span class="list-item-text">Find $I$ when&nbsp;$\mathrm{P}(x) = 1 - x$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-ii"><div class="list-item-content"><span class="list-item-text">By using the substitution $u = 1 - \sqrt{x}$, find $I$ when&nbsp;$\mathrm{P}(x) = 1$.</span><span class="inline-marks">[3]</span></div></li></ol><div class="list-item-after-content">Hence find $I$ when&nbsp;$\mathrm{P}(x) = x$.<span class="inline-marks">[2]</span></div></li></ol>

Let $I$ be the indefinite integral $\displaystyle \displaystyle\int \frac{\mathrm{P}(x)}{1-\sqrt{x}} \, \mathrm{d}x$, $0 < x < 1$, where $\mathrm{P}(x)$ is a polynomial in $x$.

2025 NJC P1 Q9

<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 the Reverse Chain Rule

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

2025 MI P1 Q3

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Differentiate $\mathrm{e}^{\cos 2x}$ with respect to $x$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find $ \displaystyle\int \mathrm{e}^{\cos 2x} \sin 4x \, \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">It is given that $2 \cos A \sin B = \sin(A + B) - \sin(A - B)$. By using this result, find $\displaystyle\int \mathrm{e}^{\cos 2x} (\cos 3x \sin x) \, \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Differentiate $\mathrm{e}^{\cos 2x}$ with respect to $x$.

Trigonometric Integration Using Product-to-Sum Formulae

<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

2025 HCI P1 Q9

<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 Splitting the Numerator

Modulus Functions

Area Bounded by Curves

Asymptotes and Discontinuities

Graphs of Modulus Function

Integration by Completing the Square

Standard Integrals with Quadratic Denominators

2025 ASRJC P2 Q5

Graphing Techniques

Transformations of Graphs

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Use the substitution $u^2 = 1 + x$ to find $\int \frac{x}{2(1+x)\sqrt{1+x}} \mathrm{d}x$.</span></div></li><li><div class="list-item-paragraph">Hence find $\int \tan^{-1} \sqrt{x+1} \mathrm{d}x$</div></li></ol>

Use the substitution $u^2 = 1 + x$ to find $\int \frac{x}{2(1+x)\sqrt{1+x}} \mathrm{d}x$.

<p>Solve $\displaystyle\int{x\ln x}\mathrm{d}x$.</p>

Solve $\displaystyle\displaystyle \displaystyle\int{x\ln x}\mathrm{d}x$.

Worked Example 4

<p>Find the integral $\displaystyle\int x \ln x$ $dx$. Hence, find $\displaystyle\int_{1}^{e^{a}} x (\ln x)^{2} dx$.</p>

Find the integral $\displaystyle\displaystyle \displaystyle\int x \ln x$ $dx$. Hence, find $\displaystyle\displaystyle \displaystyle\int_{1}^{e^{a}} x (\ln x)^{2} dx$.

<p>Find the integral $\displaystyle\int x^{n-1} e^{x^{n}} dx$. Hence, find $\displaystyle\int_{0}^{1} x^{2n-1} e^{x^{n}} dx$, in terms of $n$.</p>

Find the integral $\displaystyle\displaystyle \displaystyle\int x^{n-1} e^{x^{n}} dx$. Hence, find $\displaystyle\displaystyle \displaystyle\int_{0}^{1} x^{2n-1} e^{x^{n}} dx$, in terms of $n$.

<ol class="i-list"><li><div class="list-item-paragraph">Find $\frac{\\mathrm{d}}{\\mathrm{d}x}(\sin x^{2})$.</div></li><li><div class="list-item-paragraph">Hence, find $\displaystyle\int x^{3}\cos x^{2} dx$.</div></li></ol>

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