A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 11: Integration · Lesson 3

Integration by substitution

115 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-integration-by-substitution/

Date

Do Now

Quick questions to start.

A1
Needed today
Given $u = 2 x + 5$ , write $d x$ in terms of $d u$ .
[2 marks]
Answer
A2
Needed today
Given $u = x - 1$ , write $x$ in terms of $u$ , and $d x$ in terms of $u$ and $d u$ .
[3 marks]
Answer
A3
Common slip
An integral runs from $x = 1$ to $x = 5$ . It is rewritten with $u = 2 x - 1$ . Write down the new limits for $u$ .
[2 marks]
Answer
A4
Needed today
Find $\int (u^{\frac{3}{2}} - u^{\frac{1}{2}}) d u$ .
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Use the substitution $u = 2 x + 1$ to find $\int x (2 x + 1)^{4} d x$ .

Hint From

u = 2 x + 1

d x = \frac{1}{2} d u

and

x = \frac{u - 1}{2}

. Every

x

must go, not only the bracket.

B1
Use the substitution $u = 3 x - 1$ to find $\int x (3 x - 1)^{5} d x$ .
[5 marks]
Answer

Example 2

Use the substitution $u = x - 1$ to find the exact value of $\int_{2}^{10} \frac{x}{x - 1} d x$ .

Hint

x = u^{2} + 1

and

d x = 2 u d u

. Change the limits:

x = 2

gives

u = 1

, and

x = 10

gives

u = 3

. Then stay in

u

B2
Use the substitution $u = x - 3$ to find the exact value of $\int_{4}^{7} x x - 3 d x$ .
[6 marks]
Answer

Example 3

Use the substitution $u = e^{x}$ to find $\int \frac{e ^{2 x}}{e ^{x} + 2} d x$ .

Hint

d u = e^{x} d x

, so

e^{2 x} d x = e^{x} \times e^{x} d x = u d u

. Then

\frac{u}{u + 2} = 1 - \frac{2}{u + 2}

B3
Use the substitution $u = e^{x}$ to find the exact value of $\int_{0}^{l n 3} \frac{e ^{2 x}}{e ^{x} + 1} d x$ .
[5 marks]
Answer

Example 4

Use the substitution $u = 1 + sin x$ to find the exact value of $\int_{0}^{\frac{π}{2}} cos x 1 + sin x d x$ .

Hint

d u = cos x d x

, which is already in the integral. The limits become

u = 1

and

u = 2

B4
Use the substitution $u = 1 + sin x$ to find the exact value of $\int_{0}^{\frac{π}{2}} cos x (1 + sin x)^{3} d x$ .
[4 marks]
Answer

Example 5

Use the substitution $x = sin θ$ to find the exact value of $\int_{0}^{\frac{1}{2}} \frac{1}{1 - x ^{2}} d x$ .

Hint

d x = cos θ d θ

and

1 - sin^{2} θ = cos θ

. When

x = \frac{1}{2}

θ = \frac{π}{6}

B5
Use the substitution $x = 2 sin θ$ to find the exact value of $\int_{0}^{3} \frac{1}{4 - x ^{2}} d x$ .
[4 marks]
Answer

Practice

A-Level

C1
Zoe uses $u = x - 1$ on $\int_{1}^{2} x (x - 1)^{4} d x$ . She writes $\int_{1}^{2} (u + 1) u^{4} d u$ and gets $\frac{167}{10}$ . Explain her mistake. Then find the correct value.
[5 marks]
Answer

C2
Ali uses $u = x^{2} + 1$ on $\int x (x^{2} + 1)^{3} d x$ . He writes $\int x u^{3} d u = \frac{x u ^{4}}{4} + c$ . Explain his mistake. Then find the integral.
[4 marks]
Answer
C3
Use the substitution $u = ln x$ to find the exact value of $\int_{e}^{e^{2}} \frac{1}{x ln x} d x$ .
[4 marks]
Answer
C4
Use the substitution $u = cos x$ to find the exact value of $\int_{0}^{\frac{π}{3}} sin^{3} x d x$ .
[5 marks]
Answer
C5
Use the substitution $u = x$ to find the exact value of $\int_{1}^{9} \frac{1}{x ( 1 + x )} d x$ .
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
This question is about $\int \frac{x}{2 x + 1} d x$ .
(a)
Use the substitution $u^{2} = 2 x + 1$ to show that $\int \frac{x}{2 x + 1} d x = \frac{1}{3} (x - 1) 2 x + 1 + c$ .
[5 marks]
(b)
Hence find the exact value of $\int_{4}^{12} \frac{x}{2 x + 1} d x$ .
[2 marks]
Answer
(c)
Find the exact value of $\int_{0}^{4} \frac{x}{2 x + 1} d x$ .
[2 marks]
Answer
Total for D1: 9 marks

D2
Ben uses $u = 1 + 2 sin x$ on $\int_{0}^{\frac{π}{6}} \frac{cos x}{( 1 + 2 sin x ) ^{2}} d x$ . He writes $\frac{1}{2} \int_{0}^{\frac{π}{6}} u^{- 2} d u$ .
Ben is wrong.
(a)
Explain his mistake and write down the correct limits.
[2 marks]
(b)
Find the exact value of the integral.
[4 marks]
Answer
(c)
Use the same substitution to find the exact value of $\int_{0}^{\frac{π}{6}} \frac{sin 2 x}{( 1 + 2 sin x ) ^{2}} d x$ .
[3 marks]
Answer
Total for D2: 9 marks
D3
Use the substitution $u^{2} = e^{x} - 1$ to find the exact value of $\int_{l n 2}^{l n 5} \frac{e ^{2 x}}{e ^{x} - 1} d x$ .
[7 marks]
Answer
D4
This question is about $\int_{0}^{\frac{3}{2}} \frac{x ^{2}}{9 - x ^{2}} d x$ .
(a)
Use the substitution $x = 3 sin θ$ to show that the integral becomes $\int_{0}^{\frac{π}{6}} 9 sin^{2} θ d θ$ .
[3 marks]
(b)
Hence find the exact value of the integral. Give your answer in the form $aπ + b 3$ .
[6 marks]
Answer
Total for D4: 9 marks
D5
This question is about substituting $u = ln x$ .
(a)
Find $\int \frac{( ln x ) ^{2}}{x} d x$ .
[2 marks]
Answer
(b)
Find the exact value of $\int_{1}^{e^{2}} \frac{( ln x ) ^{2}}{x} d x$ .
[2 marks]
Answer
(c)
Find the exact value of $\int_{e}^{e^{3}} \frac{2}{x ( ln x ) ^{2}} d x$ .
[4 marks]
Answer
Total for D5: 8 marks
D6
This question is about the curve $y = x 4 - x^{2}$ for $0 \leq x \leq 2$ .
(a)
Use the substitution $u = 4 - x^{2}$ to find the exact value of $\int_{0}^{2} x 4 - x^{2} d x$ .
[4 marks]
Answer
(b)
Use the same substitution to find the exact value of $\int_{0}^{2} x^{3} 4 - x^{2} d x$ .
[5 marks]
Answer
Total for D6: 9 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Use the substitution $x = tan θ$ to find the exact value of $\int_{0}^{1} \frac{1}{( 1 + x ^{2} ) ^{2}} d x$ .
[9 marks]
Hint 1 · What to try
$d x = sec^{2} θ d θ$ and $1 + tan^{2} θ = sec^{2} θ$ . The limits are $θ = 0$ and $θ = \frac{π}{4}$ .
Hint 2 · The first line
The integrand becomes $\frac{sec ^{2} θ}{sec ^{4} θ} = cos^{2} θ$ .
Hint 3 · The full method
Use $cos^{2} θ = \frac{1}{2} + \frac{1}{2} cos 2 θ$ , then integrate and substitute.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can choose a substitution, find dx in terms of du, and write the whole integral in terms of u.
I can change the limits of a definite integral so I never have to go back to x.
I can use a substitution such as u² = 2x + 1 or u = eˣ to integrate a function I cannot spot.
I can use x = sin θ or x = tan θ to find an exact value involving π.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers