Further Integration (Cambridge International A-Level Mathematics): Exam Questions

(i) Find the integral

\int \frac{1}{x} \text{d}x

(ii) Use calculus to evaluate

\int_{0}^{1} \text{e}^{x} \text{d}x

(iii) Find an expression for $y$ given that

y = \int 3 \, \text{cos} \; \theta \, \text{d}\theta

(i) Integrate

\int \text{sin} \; (2x + 1) \, \text{d}x

(ii) Use calculus to find the exact value of

\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \text{sec}^{2} \left(2\theta - \frac{\pi}{6}\right) \text{d}\theta

(iii) Find an expression for $y$ given that

$\frac{\text{d}y}{\text{d}x} = 4 \text{cos} ( 5 x + 2 )$

(i) Integrate

\int 8(2x - 1)^{3} \, \text{d}x

(ii) Use calculus to find the exact value of

\int_{0}^{\frac{\pi}{4}} \text{sin} \; 2x \, \text{d}x

(iii) Find an expression for $y$ given that

$\frac{\text{d}y}{\text{d}x} = 3 \text{e}^{3x}$

Given that $u = 3 x + 2$, show that

(i) $\text{d} u = 3 \text{d} x$

(ii) \int 3 \, \text{cos} \; (3x + 2) \, \text{d}x = \int \text{cos} \; u \, \text{d}u

Hence find an expression in terms of $x$ for the integral

\int 3 \, \text{cos} \; (3x + 2) \, \text{d}x

Given the identity $\text{cos} 2 A \equiv 1 - 2 \text{sin}^{2} A$, show that

\text{sin}^{2} A \equiv \frac{1}{2}\left(1 - \text{cos} \; 2A\right)

Hence find the exact value of

\int_{\frac{1}{2}\pi}^{\pi} \text{sin}^{2} x \, \text{d}x

Using the substitution $u = 4 x + 1$, find the exact value of

\int_{2}^{6} 4\sqrt{4x + 1} \, \text{d}x

(i) Find an expression for $y$ given that

$\frac{\text{d}y}{\text{d}x} = 7 \text{e}^{3x-4}$

(ii) Find

\int \frac{1}{4x + 9} \, \text{d}x

(i) Given that $\text{f} ( x ) = 2 x^{2} + 5$, find $\text{f} ' ( x )$

(ii) Hence, or otherwise, find

\int \frac{4x}{2x^{2} + 5} \, \text{d}x

Use integration by parts to find

\int -3x \, \text{sin} \; x \, \text{d}x

Find the exact value of

\int_{1}^{2} \text{e}^{3x + 2} \, \text{d}x

Find

\int \frac{1}{x^{2} + 5^{2}} \, \text{d}x

(i) Show that

$\frac{4}{4x^{2}+36} = \frac{1}{x^{2}+3^{2}}$

(ii) Hence find an expression for $y$ given that

$\frac{\text{d}y}{\text{d}x} = \frac{4}{4x^{2}+36}$

Use integration by parts to show that

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

where $c$ is a constant.

(i) Find

\int \frac{1}{2 - 3x} \, \text{d}x

(ii) Find an expression for $y$ given that

y = \int \left(\frac{2}{2x + 1} - \frac{1}{3 - x}\right) \text{d}x

giving your answer as a single logarithm.

Find

\int \frac{1}{2x + 3} \, \text{d}x

Find an expression for $y$ given that

$\frac{\text{d}y}{\text{d}x} = 3 \text{e}^{3x+1}$

Find

\int \text{sin} \; x \, \text{d}x

Find the exact value of

\int_{1}^{4} \frac{1}{x} \, \text{d}x

Find an expression for $y$ given that

y = \int 7\text{e}^{7x} \, \text{d}x

Find

\int \text{cos} \; 2x \, \text{d}x

Find the value of

\int_{0}^{2} (3x - 1)^{3} \, \text{d}x

Find an expression for $y$ given that

$\frac{\text{d}y}{\text{d}x} = \text{e}^{5x}$

Using the substitution $u = 5 x - 2$, find

\int -15 \, \text{sin} \; (5x - 2) \, \text{d}x

Find an expression for $\text{f} ( x )$ given that

\text{f}(x) = \int \text{sin} \; (2x + 1) \, \text{d}x

Find an expression for $y$ given that

$\frac{\text{d}y}{\text{d}x} = 3 \text{cos} ( 3 x + 5 )$

Find the exact value of

\int_{\frac{7\pi}{6}}^{\frac{4\pi}{3}} \text{sec}^{2} \; (x - \pi) \, \text{d}x

Show that

\int_{2}^{5} 4\text{e}^{2x - 4} \, \text{d}x = 2\left(\text{e}^{6} - 1\right)

Find

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

Write down

\int 3\text{e}^{3x} \, \text{d}x

Find an expression for $\text{f} ( x )$ given that

\text{f}(x) = \int \frac{5}{2x + 3} \, \text{d}x

Find an expression for $y$ given that

$\frac{\text{d}y}{\text{d}x} = 2 \text{e}^{3x-5} - 5 \text{e}^{5x}$

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

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

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

Hence find

\int \text{f}(x) \, \text{d}x \quad \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

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

find the exact value of

\int_{\frac{1}{4}\pi}^{\frac{1}{2}\pi} \text{cos}^{2} \theta \, \text{d}\theta

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

(ii) Hence, or otherwise, find

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

Use the identity

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

to show that

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

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

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

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

Hence find

\int \text{f}(x) \, \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 $y = \text{f} ( x )$ passes through the point $( 0 , 4 )$ and that

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

Find $\text{f} ( x )$

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

Show that

\int \text{tan} \; x \, \text{d}x = \text{ln} \; |\text{sec} \; x| + 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 $u = 3 x^{2} + 1$, 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 $\text{e}^{\text{f}(x)} + \text{e}^{\text{g}(x)} + \text{e}^{\text{h}(x)}$, where $\text{f} ( x )$, $\text{g} ( x )$ and $\text{h} ( x )$ 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 $\text{f} ' ( x ) = 2 \text{tan} 3 x$ and that $\text{f} ( 0 ) = \frac{2}{3}$, show that

\text{f}(x) = \frac{2}{3}\left(1 + \text{ln} \; |\text{sec} \; 3x|\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 $a \neq 0$.

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 $a \neq 0$.

Show that, for $θ \neq k π$ (where $k$ is an integer),

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

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

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

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

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

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} \; |\text{sec} \; kx| + 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 $\text{f} ( θ )$ given that

\text{f}'(\theta) = \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}'(\theta) = \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 $\text{f} ( θ )$.

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

\text{f}(x) = \text{ln} \; \sqrt{\left|3 \, \text{sec}\left(\frac{x}{2}\right)\right|}