Further Integration (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

Using the substitution $u=\text{sin}x$, find the exact value of

${\int }_0^{\frac{1}{2}\pi }{\text{cos}}^{2}2x \text{cos}x \text{d}x$

Let

$\text{f}\left(x\right)=\frac{2x-1}{(x+1)(x-2)}$

Express $\text{f}\left(x\right)$ in partial fractions.

Hence find

$\int \text{f}\left(x\right) \text{d}x\text{for}x>2$

giving your answer as a single logarithm.

Using the substitution $u=x+4$, show that

${\int }_1^2\frac{x}{x+4} \text{d}x=1+4 \text{ln}\frac{5}{6}$

Given that

$\text{cos}2\theta \equiv 2 {\text{cos}}^2\theta -1$

find the exact value of

${\int }_{\frac{1}{4}\pi }^{\frac{1}{2}\pi }{\text{cos}}^2\theta \text{d}\theta$

(i) Given that $\text{f}\left(x\right)=2x^3+4x$, find $\text{f}\text{'}\left(x\right)$

(ii) Hence, or otherwise, find

$\int \frac{3x^2+2}{2x^3+4x} \text{d}x$

Use the identity

$\text{sin}2A\equiv 2 \text{sin}A \text{cos}A$

to show that

$4 \text{sin}\frac{\theta }{2} \text{cos}\frac{\theta }{2}\equiv 2 \text{sin}\theta$

Hence, or otherwise, find the integral

$\int 4 \text{sin}\frac{\theta }{2} \text{cos}\frac{\theta }{2} \text{d}\theta$

Use integration by parts to find, in terms of $\text{e}$, the exact value of

${\int }_0^1(5x-4){\text{e}}^{3x} \text{d}x$

Let

$\text{f}\left(x\right)=\frac{11}{(2x-3)(x+4)}$

Express $\text{f}\left(x\right)$ in partial fractions.

Hence find

$\int \text{f}\left(x\right) \text{d}x$

writing your answer as a single logarithm.

Find the value of

${\int }_2^4\frac{1}{4+9x^2} \text{d}x$

giving your answer correct to 3 significant figures.

Given that the graph of $y=\text{f}\left(x\right)$ passes through the point $(0,4)$ and that

$\text{f}\text{'}\left(x\right)=\frac{12x^2+7}{2\left(4x^3+7x+4\right)}$

Find $\text{f}\left(x\right)$

Explain why $x=-\frac{1}{2}$ must be excluded from the domain of $\text{f}\left(x\right)$

Show that

$\int \text{tan}x \text{d}x=\text{ln}\left|\text{sec}x\right|+c$

where $c$ is a constant.

Find

$\int 5\left({\text{e}}^{5x}-{\text{e}}^{-5x}\right)\text{d}x$

Find an expression for $y$ given that

$y=\int \left(\text{sin}x+\text{cos}x\right)\text{d}x$

Evaluate

${\int }_{-8}^{-2}\frac{1}{x} \text{d}x$

giving your answer as a single logarithm.

Find

$\int 2 \text{sin}x \text{cos}x \text{d}x$

Find the value of

${\int }_1^3(4x+1{)}^5 \text{d}x$

Using the substitution $u=3x^2+1$, find

$\int 8x \text{sin}(3x^2+1) \text{d}x$

Using the substitution $u=2+\text{ln}x$, show that

$\int \frac{1}{x(2+\text{ln}x{)}^3} \text{d}x=\frac{-1}{2(\text{ln}x+2{)}^2}+c$

where $c$ is the constant of integration.

Show that

${\text{sin}}^2(3x+2)\equiv \frac{1}{2}\left(1-\text{cos}(6x+4)\right)$

Hence, or otherwise, find an expression for

$\int {\text{sin}}^2(3x+2) \text{d}x$

Write ${\text{e}}^{2x}\left(1+{\text{e}}^{2x}+{\text{e}}^2\right)$ in the form ${\text{e}}^{\text{f}\left(x\right)}+{\text{e}}^{\text{g}\left(x\right)}+{\text{e}}^{\text{h}\left(x\right)}$, where $\text{f}\left(x\right)$, $\text{g}\left(x\right)$ and $\text{h}\left(x\right)$ are all linear functions of $x$.

Hence show that

${\int }_0^1{\text{e}}^{2x}\left(1+{\text{e}}^{2x}+{\text{e}}^2\right)\text{d}x=\frac{3}{4}\left({\text{e}}^4-1\right)$

Given that $\text{f}\text{'}\left(x\right)=2 \text{tan}3x$ and that $\text{f}\left(0\right)=\frac{2}{3}$, show that

$\text{f}\left(x\right)=\frac{2}{3}\left(1+\text{ln}\left|\text{sec}3x\right|\right)$

Show that

${\int }_0^1{\text{e}}^{ax+b} \text{d}x={\text{e}}^b\left(\frac{{\text{e}}^a-1}{a}\right)$

where $a$ and $b$ are constants, and $a\ne 0$.

Using your working from part (a), or otherwise, evaluate

${\int }_0^c{\text{e}}^{ax+b} \text{d}x$

giving your answer in terms of $a$, $b$ and $c$, where $a$, $b$ and $c$ are constants, and $a\ne 0$.

Show that, for $\theta \ne k\pi$ (where $k$ is an integer),

$\frac{2-2 {\text{cos}}^2\theta }{\text{sin}2\theta }=\text{tan}\theta$

Hence show that

${\int }_{\frac{1}{6}\pi }^{\frac{1}{3}\pi }\frac{2-2 {\text{cos}}^2\theta }{\text{sin}2\theta } \text{d}\theta =\frac{1}{2} \text{ln}3$

Find the exact value of

${\int }_{\frac{1}{2}\pi }^{\frac{5}{6}\pi }\frac{2 \text{cos}x}{1-\text{cos}2x} \text{d}x$

Show that

${\int }_0^1\frac{3x{\text{e}}^{-3x^2}}{3-2{\text{e}}^{-3x^2}} \text{d}x=\frac{1}{4} \text{ln}\left(3-2{\text{e}}^{-3}\right)$

Use integration by parts to find

$\int \left(2x^2-1\right){\text{e}}^x \text{d}x$

Show that

$\int \text{ln}x \text{d}x=x \text{ln}x-x+c$

where $c$ is the constant of integration.

Express

$\frac{x^2-4x+7}{(x-1)(x-3{)}^2}$

as partial fractions.

Hence, or otherwise, find

$\int \frac{x^2-4x+7}{(x-1)(x-3{)}^2} \text{d}x$

Show that

${\int }_0^{\sqrt{\frac{3}{2}}}\frac{4}{9+2x^2} \text{d}x=\frac{\pi \sqrt{2}}{9}$

Given

$\text{f}\left(x\right)=\int \frac{3\left(8x^3+3x\right)}{2\left(4x^4+3x^2+5\right)} \text{d}x$

for $x\in \mathrm{ℝ}$, and that the graph of $y=\text{f}\left(x\right)$ passes through the point $(1, \text{ln}144)$, find $\text{f}\left(x\right)$.

Find the value of

${\int }_2^6\left(\frac{1}{x}+\frac{4}{2x}+\frac{3}{3x-2}\right)\text{d}x$

giving your answer in the form $p \text{ln}p+q \text{ln}q$, where $p$ and $q$ are prime numbers to be found.

Find an expression for $y$ given that

$\frac{\text{d}y}{\text{d}x}=5 {\text{cos}}^{2}4x \text{sin}4x$

Find

$\int 3x{\left(5x^2+4\right)}^4 \text{d}x$

Find an expression for $y$ given that

$y=\int 6x^2{\text{e}}^{x^3} \text{d}x$

Find

$\int (16-32x) \text{sin}\left[(4x-2{)}^2\right]\text{d}x$

Find

$\int {\left(2 \text{tan}x+3 \text{sec}x\right)}^2\text{d}x$

Show that

$\int \text{tan}kx \text{d}x=\frac{1}{k} \text{ln}\left|\text{sec}kx\right|+c$

where $k$ is a constant, and $c$ is the constant of integration.

Find the exact value of

${\int }_{\frac{1}{18}\pi }^{\frac{1}{9}\pi }\frac{{\text{cosec}}^{2}3\theta }{3 \text{cot}3\theta } \text{d}\theta$

writing your answer in the form $a \text{ln}b$, where $a$ and $b$ are rational numbers to be found.

Show that

$\left(\text{cos}\left(\theta +\frac{\pi }{8}\right)+\text{sin}\left(\theta +\frac{\pi }{8}\right)\right)\left(\text{sin}\left(\theta +\frac{\pi }{8}\right)-\text{cos}\left(\theta +\frac{\pi }{8}\right)\right)\equiv -\text{cos}\left(2\theta +\frac{\pi }{4}\right)$

Hence, or otherwise, find the exact value of

${\int }_0^{\frac{1}{8}\pi }\left({\text{sin}}^2\left(\theta +\frac{\pi }{8}\right)-{\text{cos}}^2\left(\theta +\frac{\pi }{8}\right)\right)\text{d}\theta$

Show that

$\left(1+\text{cot}\left(2\theta +\frac{\pi }{4}\right)\right)\left(1-\text{cot}\left(2\theta +\frac{\pi }{4}\right)\right)\equiv 2-{\text{cosec}}^2\left(2\theta +\frac{\pi }{4}\right)$

Hence, or otherwise, find an expression for $\text{f}\left(\theta \right)$ given that

$\text{f}\text{'}\left(\theta \right)=\left(2+2 \text{cot}\left(2\theta +\frac{\pi }{4}\right)\right)\left(2-2 \text{cot}\left(2\theta +\frac{\pi }{4}\right)\right)$

Find

$\int x^2 \text{sin}3x \text{d}x$

Find

$\int \frac{\text{ln}x}{x^3} \text{d}x$

Find

$\int \frac{8x^2-8x-1}{\left(4x^2-1\right)(x-2)} \text{d}x$

Express

$\frac{7y+13}{12y^2+43y+36}$

in partial fractions.

Hence find

$\int \frac{7x^2+13}{12x^4+43x^2+36} \text{d}x$

Use integration by parts to show that

$\int {\text{e}}^x \text{sin}x \text{d}x=\frac{1}{2}{\text{e}}^x\left(\text{sin}x-\text{cos}x\right)+c$

where $c$ is the constant of integration.

Show that there are no positive values of $a$ and $b$ that satisfy the equation

${\int }_0^1\frac{2a}{ax+b} \text{d}x={\int }_b^{2a}\frac{1}{x} \text{d}x$

Given that

$\text{f}\text{'}\left(\theta \right)=\frac{9}{3-3 {\text{sin}}^2\left(3\theta -\frac{\pi }{6}\right)}$

and also that

${\left[\text{f}\left(\frac{\pi }{6}\right)\right]}^2-{\left[\text{f}\left(\frac{\pi }{9}\right)\right]}^2=\frac{8}{3}\left(1-\sqrt{3}\right)$

find $\text{f}\left(\theta \right)$.

Given that $\text{f}\text{'}\left(x\right)=\frac{1}{4} \text{tan}\left(\frac{x}{2}\right)$ and $\text{f}\left(\frac{2\pi }{3}\right)=\frac{1}{2} \text{ln}6$, show that