A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 11: Integration · Lesson 4

Integration by parts

114 marks

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Quick questions to start.

A1
Needed today
Differentiate $x e^{2 x}$ with respect to $x$ .
[2 marks]
Answer
A2
Needed today
Find $\int e^{3 x} d x$ .
[2 marks]
Answer
A3
Needed today
Differentiate $x ln x$ with respect to $x$ .
[2 marks]
Answer
A4
Common slip
Find $\int cos 2 x d x$ . Check your answer by differentiating it.
[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

Find $\int x e^{x} d x$ .

Hint Take

u = x

, because it becomes

1

when you differentiate it, and

\frac{d v}{d x} = e^{x}

, so

v = e^{x}

B1
Find $\int x e^{3 x} d x$ .
[4 marks]
Answer

Example 2

Find $\int x cos x d x$ .

Hint Take

u = x

and

\frac{d v}{d x} = cos x

, so

v = sin x

. Watch the sign when you integrate

sin x

B2
Find $\int x sin 2 x d x$ .
[4 marks]
Answer

Example 3

Find $\int ln x d x$ .

Hint Write it as

\int 1 \times ln x d x

. Take

u = ln x

, because you cannot integrate it yet, and

\frac{d v}{d x} = 1

, so

v = x

B3
Find the exact value of $\int_{1}^{e} x^{2} ln x d x$ .
[5 marks]
Answer

Example 4

Find $\int x^{2} e^{x} d x$ .

Hint Use parts with

u = x^{2}

. The new integral

\int 2 x e^{x} d x

needs parts again.

B4
Find $\int x^{2} cos x d x$ .
[5 marks]
Answer

Example 5

Find $\int e^{x} sin x d x$ .

Hint Call it

I

. Use parts twice with

u

as the trig function each time. The second time,

I

appears again: collect it on one side.

B5
Find the exact value of $\int_{0}^{\frac{π}{2}} e^{x} cos x d x$ .
[6 marks]
Answer

Practice

A-Level

C1
Kai finds $\int_{0}^{1} x e^{2 x} d x$ by parts. He takes $u = e^{2 x}$ and $\frac{d v}{d x} = x$ , and gets $\frac{1}{2} x^{2} e^{2 x} - \int x^{2} e^{2 x} d x$ . Explain why this does not help. Then find the exact value.
[5 marks]
Answer

C2
Ella writes $\int x sin x d x = - x cos x - \int cos x d x = - x cos x - sin x + c$ . Find her mistake, then write the correct answer.
[4 marks]
Answer
C3
Find the exact value of $\int_{1}^{2} ln 3 x d x$ . Give your answer in the form $ln a - b$ .
[5 marks]
Answer
C4
Find $\int x sec^{2} x d x$ .
[4 marks]
Answer
C5
Use integration by parts to find the exact value of $\int_{0}^{1} x (x + 1)^{3} d x$ .
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
This question is about $\int x e^{- 2 x} d x$ .
(a)
Find $\int x e^{- 2 x} d x$ .
[4 marks]
Answer
(b)
Find the exact value of $\int_{0}^{l n 2} x e^{- 2 x} d x$ .
[3 marks]
Answer
(c)
Show that $\int_{0}^{k} x e^{- 2 x} d x < \frac{1}{4}$ for every $k > 0$ .
[2 marks]
Total for D1: 9 marks

D2
Sam writes $\int x^{2} e^{2 x} d x = \frac{1}{2} x^{2} e^{2 x} + \int x e^{2 x} d x$ .
Sam is wrong.
(a)
Explain his mistake.
[2 marks]
(b)
Find $\int x^{2} e^{2 x} d x$ .
[4 marks]
Answer
(c)
Hence find the exact value of $\int_{0}^{1} x^{2} e^{2 x} d x$ .
[3 marks]
Answer
Total for D2: 9 marks
D3
Find the exact value of $\int_{1}^{e} \frac{ln x}{x ^{2}} d x$ .
[7 marks]
Answer
D4
This question is about $I = \int e^{2 x} sin x d x$ .
(a)
Use integration by parts twice to show that $I = \frac{1}{5} e^{2 x} (2 sin x - cos x) + c$ .
[6 marks]
(b)
Find the exact value of $\int_{0}^{π} e^{2 x} sin x d x$ .
[3 marks]
Answer
Total for D4: 9 marks
D5
This question is about $\int_{1}^{4} e^{x} d x$ .
(a)
Use the substitution $u = x$ to show that $\int_{1}^{4} e^{x} d x = \int_{1}^{2} 2 u e^{u} d u$ .
[3 marks]
(b)
Hence find the exact value of $\int_{1}^{4} e^{x} d x$ .
[6 marks]
Answer
Total for D5: 9 marks
D6
This question is about $\int x ln x d x$ .
(a)
Find $\int x ln x d x$ .
[3 marks]
Answer
(b)
Find the exact value of $\int_{1}^{3} x ln x d x$ .
[2 marks]
Answer
(c)
Use your answer to part (a) to find $\int x (ln x)^{2} d x$ .
[3 marks]
Answer
Total for D6: 8 marks

Extension

Stretch

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

E1
Find the exact value of $\int_{1}^{e} (ln x)^{3} d x$ .
[9 marks]
Hint 1 · What to try
Let $I_{n} = \int_{1}^{e} (ln x)^{n} d x$ . Use parts with $u = (ln x)^{n}$ and $v = x$ .
Hint 2 · The first line
You should get $I_{n} = e - n I_{n - 1}$ , because $ln e = 1$ and $ln 1 = 0$ .
Hint 3 · The full method
Start from $I_{0} = \int_{1}^{e} 1 d x = e - 1$ and work up to $I_{3}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can choose u so that it gets simpler when I differentiate it, and use the parts formula with the minus sign.
I can integrate ln x, x² ln x and similar by taking u = ln x.
I can use integration by parts twice, and spot when the integral comes back to the start.
I can find an exact value with integration by parts and the limits.

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