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iGCSE (2021 Edition)

12.02 The reverse chain rule

Worksheet
The reverse chain rule
1

Integrate the following to find an expression for y:

a
\dfrac{dy}{dx} = \left( 3 x + 5\right)^{4}
b
\dfrac{d y}{d x} = \left( 6 x - 5\right)^{2}
c
\dfrac{dy}{dx} = \dfrac{1}{\left( - 4 x + 3\right)^{4}}
d
\dfrac{dy}{dx} = \sqrt{ 2 x - 9}
2

Evaluate the following integrals:

a
\int \left(x - 5\right)^{4}\, dx
b
\int \left( 2 x + 3\right)^{3}\, dx
c
\int 6 \left(x + 2\right)^{2}\, dx
d
\int \left( 4 x - 3\right)^{ - 4 }\, dx
e
\int \dfrac{4}{\left(x - 5\right)^{3}}\, dx
f
\int \dfrac{1}{\left(x + 6\right)^{3}}\, dx
g
\int \dfrac{1}{\left( 2 x - 3\right)^{4}}\, dx
h
\int \dfrac{1}{\left( 4 x + 3\right)^{2}}\, dx
i
\int \left( 2 x - 5\right)^{\frac{1}{2}}\, dx
j
\int \left( 4 x + 5\right)^{ - \frac{1}{2} }\, dx
k
\int \left( 4 x + 5\right)^{\frac{5}{4}}\, dx
l
\int \left(1 - x\right)^{6}\, dx
m
\int 4 \left(5 - x\right)^{3}\, dx
n
\int \left(4 - 3 x\right)^{6}\, dx
o
\int 4 \left(2 - 3 x\right)^{3}\, dx
p
\int \left(4 - 3 x\right)^{ - 5 }\, dx
q
\int \dfrac{5}{\left(5 - 4 x\right)^{3}}\, dx
r
\int 4 \left( 2 x + 5\right)^{3}\, dx
s
\int x \left( 4 x + 7\right) \left( 7 x + 2\right)\, dx
t
\int \left(5 + \left( 4 x + 1\right)^{3}\right)\, dx
u
\int \left( 3 x - 2\right) \left(x + 6\right)\, dx
v
\int 8 x^{7} \left(x^{8} - 9\right)^{6}\, dx
w
\int \left( 2 x + 1\right) \left(x^{2} + x\right)^{5}\, dx
3

Evaluate the following integrals:

a
\int \sqrt{7 - 4 x}\, dx
b
\int \dfrac{5}{\sqrt{1 - 4 x}}\, dx
c
\int 3 \sqrt{ 2 x + 3}\, dx
d
\int 16 \sqrt[3]{ 6 x + 5}\, dx
e
\int \left(\sqrt{x} - 2\right)^{2}\, dx
f
\int \left(\sqrt{x - 2} + \sqrt{x + 3}\right)\, dx
g
\int \dfrac{5}{\sqrt{x - 2}}\, dx
h
\int \left(\sqrt{x} + \dfrac{5}{x}\right)^{2}\, dx
i

\int 5x\,(2x^{2}+3)^{2}\, dx

j

\int 3x^2\,(x^{3}+1)^{5}\, dx

k

\int 2x^{3}\,(2+8x^{4})^{2}\, dx

l

\int 2x\,(5+3x^{2})^{3}\, dx

m

\int x\,(2-x^{2})^{2}\, dx

n

\int 2x^{2}\,\sqrt{x^{3}-1}\, dx

o

\int 5x\,\sqrt{3x^{2}+2}\, dx

p

\int \dfrac{4x}{\sqrt{x^{2}-4}}\, dx

q

\int \dfrac{x-6}{\sqrt{x^{2}-12x+5}}\, dx

r

\int \dfrac{4x}{({x^{2}+3})^{3}}\, dx

4

Find:

a

\int 10x^{2} (x^{3}-1)^{4}\, dx

b

\int \dfrac{7x}{\sqrt{x^{2}+4}}\, dx

c

\int \dfrac{5(\sqrt{x}-1)^{2}}{3\sqrt{x}}\, dx

d

\int 28(9x^{2}+12x)(x^{3}+2x^{2}-3)^{3}\, dx

e

\int (12x-12)(7-x^{2}+2x)^{5}\, dx

Applications
5

Consider the function y=x^{5} + x^{4}.

a

Find \dfrac{d}{dx} \left(x^{5} + x^{4}\right).

b

Hence, determine \int \left( 35 x^{4} + 28 x^{3}\right)\, dx.

6

Consider the function y=x^{ - 4 }.

a

Find \dfrac{d}{dx} \left(x^{ - 4 }\right).

b

Hence, determine \int 16 x^{ - 5 }\, dx.

7

Consider the function f(x) = x^{6}.

a

Find \dfrac{d}{dx} \left(x^{6}\right).

b

Hence, determine \int 24 x^{5}\, dx.

8

Consider the function f(x) = x^{4}.

a

Find \dfrac{d}{dx} \left(x^{4}\right).

b

Hence, determine \int 4 x^{3}\, dx.

9

Consider the function y = x^{8} + x^{6}.

a

Find \dfrac{d}{dx} \left(x^{8} + x^{6}\right).

b

Hence, determine \int \left( 4 x^{7} + 3 x^{5}\right)\, dx.

10

Consider the function f(x) = \sqrt[5]{x^{6}}.

a

Find \dfrac{d}{dx} \left(\sqrt[5]{x^{6}}\right).

b

Hence, determine \int 6 \sqrt[5]{x}\, dx.

11

Consider the function y = \left(x^{2} - 3\right)^{5}.

a

Find \dfrac{d}{dx} \left(x^{2} - 3\right)^{5}.

b

Hence, determine \int 30 x \left(x^{2} - 3\right)^{4}\, dx.

12

Consider the function f(x) = \left( 5 x^{2} + 10 x - 3\right)^{5}.

a

Find \dfrac{d}{dx} \left( 5 x^{2} + 10 x - 3\right)^{5}.

b

Hence, find \int 150 \left(x + 1\right) \left( 5 x^{2} + 10 x - 3\right)^{4}\, dx.

13

Consider the function y = \left(x^{2} - 3 x + 6\right)^{5}.

a

Find \dfrac{d}{dx} \left(x^{2} - 3 x + 6\right)^{5}.

b

Hence, find \int \left(3 - 2 x\right) \left(x^{2} - 3 x + 6\right)^{4}\, dx.

14

Consider the function f(x) = \sqrt{ 4 x + 11}.

a

Find \dfrac{d}{dx} \left(\sqrt{ 4 x + 11}\right).

b

Hence, determine \int \dfrac{12}{\sqrt{ 4 x + 11}}\, dx.

15

Consider the function y = \dfrac{1}{\left(x^{2} + 7\right)^{3}}.

a

Find \dfrac{d}{dx} \left(\dfrac{1}{\left(x^{2} + 7\right)^{3}}\right).

b

Hence, determine \int - \dfrac{24 x}{\left(x^{2} + 7\right)^{4}} \, dx.

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Outcomes

0606C14.9A

Integrate functions of the form (ax + b)^n for any rational n.

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