Further Integration (Edexcel IAL Maths): Exam Questions

(i) Find the integral

$\int \frac{1}{x} \mathrm{d} x$

(ii) Use calculus to evaluate $\int_{0}^{1} e^{x} \mathrm{d} x.$

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

$y = \int 3 \cos \\theta \mathrm{d} \\theta .$

(i) Integrate

$\int 8 ( 2 x - 1 )^{3} \mathrm{d} x.$

(ii) Use calculus to find the exact value of

`integral _ 0 ^ pi / 4 end ^ sin 2 x d x`.

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

$\frac{dy}{dx}=3e^{3x}$.

Given the identity $\mathrm{cos}2A\equiv1-2\mathrm{sin}^{2}A$, show that

$\sin ^{2} A = 1 \frac{1}{2} ( 1 - \cos 2 A ).$

Hence find the exact value of

integral _ pi / 2 end _ ^ pi sin ^2 x d x.

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

(ii) Hence, or otherwise, find
$\int \frac{4 x}{2 x^{2} + 5} \mathrm{d} x.$

Find the exact value of

$\int_{1}^{2} e^{3 x + 2} \mathrm{d} x.$

(i) Integrate $\int \sin ( 2 x + 1 ) d x.$

(ii) Use calculus to find the exact value of integral _ pi / 6 end _ ^ pi / 4 end ^ sec ^2 ( 2 theta - pi / 6 ) d theta.

(iii) Find an expression for $y$ given that
$\frac{ \mathrm{d} y}{ \mathrm{d} x} = 4 \cos ( 5 x + 2 ).$

(i) Find an expression for $y$ given that
$\frac{dy}{dx}=7e^{3x-4}$.

(ii) Integrate

$\int \frac{1}{4 x + 9} \mathrm{d} x$.

(i) Integrate

$\int \frac{1}{2 - 3 x} \mathrm{d} x$.

(ii) Find an expression for $y$ given that $y = \int ( \frac{2}{2 x + 1} - \frac{1}{3 - x} ) \mathrm{d} x$

giving your answer as a single logarithm.

Find the integral

$\int \sin x \mathrm{d} x$

Use calculus to evaluate

$\int_{1}^{4} \frac{1}{x} \mathrm{d} x$

Find an expression for $y$ given that

$y = \int 7 e^{7 x} \mathrm{d} x$

Integrate

$\int \cos 2 x \mathrm{d} x$

Use calculus to find the value of

$\int_{0}^{2} ( 3 x - 1 )^{3} \mathrm{d} x$

Find an expression for $y$ given that

$\frac{dy}{dx}=e^{5x}$

Given that

$\mathrm{cos}2θ\equiv2\mathrm{cos}^{2}θ-1$

use calculus to find the exact value of

integral _ pi / 4 end _ ^ pi / 2 end ^ cos ^2 theta d theta

Find

$\int \sqrt{1 + \cot ^{2} x} \mathrm{d} x$

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

(ii) Hence, or otherwise, find

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

Integrate

$\int \frac{1}{2 x + 3} \mathrm{d} x$

Find an expression for given that

$\frac{dy}{dx}=3e^{3x+1}$

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

$\mathrm{f} ( x ) = \int \sin ( 2 x + 1 ) \mathrm{d} x$

Find an expression for given that

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

Use calculus to find the exact value of

$\int_{\frac{7 \mathrm{\pi } }{6}}^{\frac{4 \mathrm{\pi } }{3}} \sec ^{2} ( x - \mathrm{\pi } ) \mathrm{d} x$

Use the identity

$\mathrm{sin}2A\equiv2\mathrm{sin}A\mathrm{cos}A$

to show that

$4\mathrm{sin}\frac{θ}{2}\mathrm{cos}\frac{θ}{2}\equiv2\mathrm{sin}θ$

Hence, or otherwise, find the integral

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

Show that

$\int_{2}^{5} 4 e^{2 x - 4} \mathrm{d} x = 2 ( e^{6} - 1 )$

Given that the graph of $y=f(x)$ passes through the point $(0,4)$ and that

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

Find $f(x)$

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

Show that

$\int \tan x \mathrm{d} x = \ln \vert s e c x \vert + c$

where $c$ is a constant.

Find the integral

$\int 5 ( e^{5 x} - e^{- 5 x} ) \mathrm{d} x$

Find an expression for $y$ given that

$y = \int ( \sin x + \cos x ) \mathrm{d} x$

Use calculus to evaluate

$\int_{- 8}^{- 2} \frac{1}{x} \mathrm{d} x$

giving your answer as a single logarithm.

Integrate

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

Use calculus to find the value of

$\int_{1}^{3} ( 4 x + 1 )^{5} \mathrm{d} x$

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

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

Use calculus and your result from part [a] to show that

integral _ pi / 6 end _ ^ pi / 3 end ^ (2 - 2 cos ^ 2 theta)/(sin 2 theta) d theta = 1 half In 3

Use calculus to find the exact value of

integral _ pi / 2 end _ ^ (5 pi)/(6) end ^ (2 cos x)/(1 - cos 2 x) d x.

Use calculus to show that

$\int_{0}^{1} \frac{3 x e^{- 3 x^{2}} }{3 - 2 e^{- 3 x^{2}} } \mathrm{d} x = 1 \frac{1}{4} \text{In} ( 3 - 2 e^{- 3} )$.

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

$\mathrm{f} ( x ) = \int \frac{5}{2 x + 3} \mathrm{d} x$

Find an expression for $y$ given that

$\frac{dy}{dx}=2e^{3x-5}-5e^{5x}$

Show that

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

Hence, or otherwise, find an expression for

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

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

Hence show that

$\int_{0}^{1} e^{2 x} ( 1 + e^{2 x} + e^{2} ) \mathrm{d} x = \frac{3}{4} ( e^{4} - 1 )$

Given that $f^{'}(x)=2\mathrm{tan}3x$and that $\mathrm{f} ( 0 ) = \frac{2}{3}$ show that

$\mathrm{f} ( x ) = \frac{2}{3} ( 1 + \ln \vert \sec 3 x \vert )$

Given

$\mathrm{f} ( x ) = \int \frac{3 ( 8 x^{3} + 3 x )}{2 ( 4 x^{4} + 3 x^{2} + 5 )} \mathrm{d} x x \in \mathbb{R}$

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

Find an expression for $y$ given that

$\frac{dy}{dx}=5\mathrm{cos}^{2}4x\mathrm{sin}4x$

Integrate

$\int 3 x ( 5 x^{2} + 4 )^{4} \mathrm{d} x$

Find

$\int ( 2 \tan x + 3 \sec x )^{2 } \mathrm{d} x$

Show that

$\int \tan k x \mathrm{d} x = \frac{1}{k } \ln \vert \sec k x \vert + c$

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

Use calculus to find the exact value of

integral _ pi / 18 end _ ^ pi / 9 end ^ (cosec ^ 2 3 theta)/(3 cot 3 theta) d theta

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

Use calculus to find the value of

$\int_{2}^{6} ( \frac{1}{x} + \frac{4}{2 x} + \frac{3}{3 x - 2} ) \mathrm{d} x$

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

Show that

( cos ( theta + pi / 8 ) + sin ( theta + pi / 8 ) ) ( sin ( theta + pi / 8 ) - cos ( theta + pi / 8 ) ) === - cos ( 2 theta + pi / 4 )

Hence, or otherwise, find the exact value of

integral _ 0 ^ pi / 8 end ^ ( sin ^2 ( theta + pi / 8 ) - cos ^ 2 ( theta + pi / 8 ) ) d theta

Show that

$\int_{0}^{1} e^{a x + b} \mathrm{d} x = e^{b} ( \frac{e^{a} - 1}{a} )$

where $a$ and $b$ are constants, and $a\neq0$.

Using your working from part [a], or otherwise, evaluate

$\int_{0}^{c} e^{a x + b} \mathrm{d} x$

giving your answer in terms of $a,b\mathrm{and}c$, where $a,b\mathrm{and}c$ are constants, and $a\neq0$.

Show that

( 1 + cot ( 2 theta + pi / 4 ) ) ( 1 - cot ( 2 theta + pi / 4 ) ) === 2 - cosec ^2 ( 2 theta + pi / 4 )

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

f ' ( theta ) = ( 2 + 2 cot ( 2 theta + pi / 4 ) ) ( 2 - 2 cot ( 2 theta + pi / 4 ) )

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

$\int_{0}^{1 \frac{2 a}{a x + b} \mathrm{d} x = \int_{b}^{2} a} \frac{1}{x} \mathrm{d} x$

Given that

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

and also that

[ f ( pi / 6 ) ] ^2 - [ f ( pi / 9 ) ] ^2 = 8 / 3 ( 1 - sqrt( 3 )

find $\mathrm{f} ( \\theta )$.

Given that f ' ( x ) = 1 fourth tan ( x / 2 ) and f ( (2 pi)/(3) ) = 1 half In 6, show that

f ( x ) = In sqrt( | 3 sec ( x / 2 ) | )