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PSLE Foundation Mathematics

Find $\dfrac{d}{dx}\left(\dfrac{1}{2e^x+1}\right)$.

Hence find $\displaystyle \int \dfrac{e^x}{(2e^x+1)^2}\, dx$

Integration as an Inverse Process of Differentiation

Derivatives of Exponential Functions

Chain Rule

2025 TKSS PRELIMS P1 Q1

Integration

Differentiation

<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 by the Reverse Chain Rule

2016 DHS C1BT Q2

Techniques of Differentiation

Techniques of Integration

<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}}{\mathrm{d}x}\left[(2x)(3 - 5x)^{\frac{6}{5}}\right] = (6 - 22x)(3 - 5x)^{\frac{1}{5}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Hence, evaluate $\int x(3 - 5x)^{\frac{1}{5}} dx$.<span class="inline-marks">[3]</span></div></li></ol>

Product Rule

2024 GESS PRELIMS P2 Q11

<p>Given that&nbsp;$y = \frac{3\ln 2x}{x^2}$ for&nbsp;$x &gt; 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-1781506047585-uk90uikz0"><div class="list-item-paragraph"> Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{3}{x^3}(1 - 2\ln 2x)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1781506048125-wavvo5cvu"><div class="list-item-paragraph">Hence, find $\int \frac{\ln 2x}{x^3} \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Given that $y = \frac{3\ln 2x}{x^2}$ for $x > 0$.

Derivatives of Natural Logarithmic Functions

Quotient Rule

2024 FMSS PRELIMS P2 Q3

<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}}{\mathrm{d}x} \left(\frac{\ln 2x}{3x^2} \right) = \frac{1}{3x^3} - \frac{2\ln 2x}{3x^3}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Integrate $\frac{\ln 2x}{x^3}$ with respect to $x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Given that the curve $y = \mathrm{f}(x)$ passes through the point $ \left(\frac{1}{2}, 1 \right)$ and is such that&nbsp;$\mathrm{f}'(x) = \frac{\ln 2x}{x^3}$, find $\mathrm{f}(x)$.<span class="inline-marks">[3]</span></div></li></ol>

Finding the Equation of the Curve

2024 CGSS P1 Q12

<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 by Parts

Integration by Substitution

2024 VJC Promo Q10

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}}{\mathrm{d}x}\left[\left(x^2+1\right)e^{2x-1}\right]=2e^{2x-1}\left(x^2+x+1\right)$.<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-paragraph">Hence, show that&nbsp;$\int -2e^{2x-1}\left(x^2+x\right) \mathrm{d}x = -x^2e^{2x-1}+c$.<span class="inline-marks">[4]</span></div></li></ol>

Show that $\frac{\mathrm{d}}{\mathrm{d}x}\left[\left(x^2+1\right)e^{2x-1}\right]=2e^{2x-1}\left(x^2+x+1\right)$.

Hence, show that $\int -2e^{2x-1}\left(x^2+x\right) \mathrm{d}x = -x^2e^{2x-1}+c$.

TKSS 4AM 2022 Prelims P2 Q8

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$y = \mathrm{e}^x (1 + \cos x)$, show $\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{e}^x (1 + \cos x - \sin x)$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find $\int \mathrm{e}^x (\cos x - \sin x) \, \mathrm{d}x$.</span><span class="inline-marks">[3]</span></div></li></ol>

Given that $y=e^x(1+\cos x)$, show $\dfrac{dy}{dx}=e^x(1+\cos x-\sin x)$.

Hence find $\displaystyle\int e^x(\cos x-\sin x)\,dx$.

2025 CGSS PRELIMS P1 Q12

<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

<p>A particle moves in a straight line with a velocity, $v$ m s$^{-1}$. Its velocity can be modelled by the equation $v = \sin t + t \cos t$. Given the initial conditions $t = 0$, $s = 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Differentiate $s = t \sin t$ with respect to $t$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence find an expression for the displacement, $s$ at time $t$</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find an expression for the acceleration, $a$, at time $t$.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Determine whether the motion is periodic, and justify your answer.</div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">What happens to the amplitude of the motion as time progresses?</div></li></ol>

A particle moves in a straight line with a velocity, $v$ m s$^{-1}$. Its velocity can be modelled by the equation $v = \sin t + t \cos t$. Given the initial conditions $t = 0$, $s = 0$.

Differentiate $s = t \sin t$ with respect to $t$.

Hence find an expression for the displacement, $s$ at time $t$

Find an expression for the acceleration, $a$, at time $t$.

Determine whether the motion is periodic, and justify your answer.

What happens to the amplitude of the motion as time progresses?

Interpreting Motion Models

Finding the Equation of Displacement from Velocity

Differentiating Displacement and Velocity

Kinematics

<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

<p>Find $\frac{\mathrm{d}}{\mathrm{d}x}\left( \frac{\cos^5 2x}{10} \right)$ and hence, find $\displaystyle\displaystyle \displaystyle\int (x - \sin 2x \cos^4 2x) \, \mathrm{d}x$.</p>

Find $\frac{\mathrm{d}}{\mathrm{d}x}\left( \frac{\cos^5 2x}{10} \right)$ and hence, find $\displaystyle\displaystyle \displaystyle\int (x - \sin 2x \cos^4 2x) \, \mathrm{d}x$.

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express $\frac{27x^2-6x+9}{(4x-1)(x^2+3)}$ in partial fractions.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Differentiate $\ln(x^2+3)$ with respect of $x$.</span></div></li><li><div class="list-item-paragraph">Using the results of <strong>(a)</strong> and <strong>(b)</strong>, find $\displaystyle\int \frac{27x^2-6x+9}{(4x-1)(x^2+3)} \, \mathrm{d}x$.</div></li></ol>

Express $\frac{27x^2-6x+9}{(4x-1)(x^2+3)}$ in partial fractions.

Differentiate $\ln(x^2+3)$ with respect of $x$.

Using the results of **(a)** and **(b)**, find $\displaystyle\displaystyle \displaystyle\int \frac{27x^2-6x+9}{(4x-1)(x^2+3)} \, \mathrm{d}x$.

Integration Using Partial Fractions

Partial Fractions

Polynomials, Cubic Equations and Partial Fractions

<p>Given that $\frac{6+5x-8x^2}{(2x-1)(x^2+3)} = \frac{A}{2x-1} + \frac{Bx+C}{x^2+3}$, where $A$, $B$ and $C$ are constants</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">find the values of $A$, $B$ and $C$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">differentiate $\ln(x^2 + 3)$ with respect to $x$.</span></div></li><li><div class="list-item-paragraph">Using the results from parts <strong>(a)</strong> and <strong>(b)</strong>, find $\displaystyle\int \frac{6+5x-8x^2}{(2x-1)(x^2+3)} \, \mathrm{d}x$.</div></li></ol>

Given that $\frac{6+5x-8x^2}{(2x-1)(x^2+3)} = \frac{A}{2x-1} + \frac{Bx+C}{x^2+3}$, where $A$, $B$ and $C$ are constants

find the values of $A$, $B$ and $C$,

differentiate $\ln(x^2 + 3)$ with respect to $x$.

Using the results from parts **(a)** and **(b)**, find $\displaystyle\displaystyle \displaystyle\int \frac{6+5x-8x^2}{(2x-1)(x^2+3)} \, \mathrm{d}x$.

<p>Find $\frac{\mathrm{d}}{\mathrm{d}x}(\sec x)$ and hence find $\displaystyle\displaystyle \displaystyle\int \tan x \sec x \, \mathrm{d}x.$</p>

Find $\frac{\mathrm{d}}{\mathrm{d}x}(\sec x)$ and hence find $\displaystyle\displaystyle \displaystyle\int \tan x \sec x \, \mathrm{d}x.$

Derivatives of Reciprocal Trigonometric Functions

<p>Given that&nbsp;$y = 2e^x (\sin x - \cos x)$, show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = 4e^x \sin x$. Hence, find $\displaystyle\displaystyle \displaystyle\int e^x \sin x \, \mathrm{d}x.$</p>

Given that $y = 2e^x (\sin x - \cos x)$, show that $\frac{\mathrm{d}y}{\mathrm{d}x} = 4e^x \sin x$. Hence, find $\displaystyle\displaystyle \displaystyle\int e^x \sin x \, \mathrm{d}x.$

<p>Find $\frac{\mathrm{d}}{\mathrm{d}x}(x^2\ln 3x)$. Hence, find $\displaystyle\displaystyle \displaystyle\int x\ln 3x \, \mathrm{d}x.$</p>

Find $\frac{\mathrm{d}}{\mathrm{d}x}(x^2\ln 3x)$. Hence, find $\displaystyle\displaystyle \displaystyle\int x\ln 3x \, \mathrm{d}x.$

<p>Differentiate $y = 5xe^{2x+1}$ with respect to $x$ and hence, find $\displaystyle\int xe^{2x+1} \, \mathrm{d}x.$</p>

Differentiate $y = 5xe^{2x+1}$ with respect to $x$ and hence, find $\displaystyle\displaystyle \displaystyle\int xe^{2x+1} \, \mathrm{d}x.$

<p>Given that $y = 2xe^{\frac{x}{2}}$, find $\frac{\mathrm{d}y}{\mathrm{d}x}$. Hence, find $\displaystyle\int xe^{\frac{x}{2}} \, \mathrm{d}x.$</p>

Given that $y = 2xe^{\frac{x}{2}}$, find $\frac{\mathrm{d}y}{\mathrm{d}x}$. Hence, find $\displaystyle\displaystyle \displaystyle\int xe^{\frac{x}{2}} \, \mathrm{d}x.$

<p>Differentiate $\frac{2}{2-x^2}$ with respect to $x$. Hence find $\displaystyle\displaystyle \displaystyle\int \frac{x}{(2-x^2)^2} \, \mathrm{d}x.$</p>

Differentiate $\frac{2}{2-x^2}$ with respect to $x$. Hence find $\displaystyle\displaystyle \displaystyle\int \frac{x}{(2-x^2)^2} \, \mathrm{d}x.$

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