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

Trigonometric identities and the reverse chain rule

113 marks

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Date

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Quick questions to start.

A1
Needed today
Use $cos 2 x = 1 - 2 sin^{2} x$ to write $sin^{2} x$ in terms of $cos 2 x$ .
[2 marks]
Answer
A2
Needed today
Differentiate $ln (x^{2} + 5)$ .
[2 marks]
Answer
A3
Needed today
Differentiate $(x^{3} + 1)^{5}$ .
[2 marks]
Answer
A4
Common slip
Write $tan^{2} x$ in terms of $sec^{2} x$ . Then find $\int tan^{2} x d x$ .
[3 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 $\int cos^{2} x d x$ .

Hint There is no rule for a square of

cos x

. Use

cos 2 x = 2 cos^{2} x - 1

to write

cos^{2} x = \frac{1}{2} + \frac{1}{2} cos 2 x

, then integrate each term.

B1
Find $\int sin^{2} 3 x d x$ .
[5 marks]
Answer

Example 2

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

Hint Divide

sin^{2} x + cos^{2} x = 1

sin^{2} x

1 + cot^{2} x = cosec^{2} x

. And

\int cosec^{2} x d x = - cot x

B2
Find $\int cot^{2} 2 x d x$ .
[4 marks]
Answer

Example 3

Find $\int \frac{6 x ^{2}}{x ^{3} + 2} d x$ .

Hint The derivative of the bottom,

x^{3} + 2

, is

3 x^{2}

. The top is

2 \times 3 x^{2}

, so the integral is

2 ln ∣ x^{3} + 2∣

. Check by differentiating.

B3
Find the exact value of $\int_{0}^{\frac{π}{2}} \frac{cos x}{3 + 2 sin x} d x$ .
[5 marks]
Answer

Example 4

Find $\int x (x^{2} + 1)^{3} d x$ .

Hint Try

(x^{2} + 1)^{4}

. Its derivative is

4 (x^{2} + 1)^{3} \times 2 x = 8 x (x^{2} + 1)^{3}

, which is

8

times too big.

B4
Find $\int sin x cos^{4} x d x$ .
[4 marks]
Answer

Example 5

Find $\int tan x d x$ .

Hint Write

tan x = \frac{sin x}{cos x}

. The top is minus the derivative of the bottom.

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

Practice

A-Level

C1
Raj writes: " $\int sin^{2} x d x = \frac{1}{3} sin^{3} x + c$ ." Explain his mistake. Then find $\int sin^{2} x d x$ .
[4 marks]
Answer

C2
Mia writes: " $\int \frac{1}{x ^{2} + 4} d x = ln (x^{2} + 4) + c$ ." Show, by differentiating her answer, that she is wrong. Then find $\int \frac{x}{x ^{2} + 4} d x$ .
[5 marks]
Answer
C3
Find the exact value of $\int_{0}^{\frac{π}{4}} sin 2 x cos 2 x d x$ .
[4 marks]
Answer
C4
Find the exact value of $\int_{0}^{l n 2} \frac{e ^{2 x}}{1 + e ^{2 x}} d x$ . Give your answer as a single logarithm.
[5 marks]
Answer
C5
Find $\int (sin x + cos x)^{2} d x$ .
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
This question is about $cos^{2} 2 x$ .
(a)
Show that $cos^{2} 2 x = \frac{1}{2} + \frac{1}{2} cos 4 x$ .
[2 marks]
(b)
Hence find $\int cos^{2} 2 x d x$ .
[2 marks]
Answer
(c)
Find the exact value of $\int_{0}^{\frac{π}{8}} (cos 2 x + sin 2 x)^{2} d x$ .
[5 marks]
Answer
Total for D1: 9 marks

D2
Tom writes: "The derivative of $x^{2} + 1$ is in the top, so $\int \frac{4 x}{x ^{2} + 1} d x = 4 ln (x^{2} + 1) + c$ ."
Tom is wrong.
(a)
Explain his mistake.
[2 marks]
(b)
Find the exact value of $\int_{0}^{2} \frac{4 x}{x ^{2} + 1} d x$ .
[3 marks]
Answer
(c)
Find the exact value of $\int_{1}^{3} \frac{4 x ^{3}}{x ^{4} + 1} d x$ , giving your answer in the form $ln k$ .
[4 marks]
Answer
Total for D2: 9 marks
D3
Find the exact value of $\int_{0}^{\frac{π}{3}} (2 tan^{2} x + sin^{2} x) d x$ . Give your answer in the form $a 3 + bπ$ , where $a$ and $b$ are rational.
[8 marks]
Answer
D4
This question is about $e^{c o s 2 x}$ .
(a)
Find $\frac{d}{d x} (e^{c o s 2 x})$ .
[2 marks]
Answer
(b)
Hence find $\int sin x cos x e^{c o s 2 x} d x$ .
[3 marks]
Answer
(c)
Find the exact value of $\int_{0}^{\frac{π}{2}} sin x cos x e^{c o s 2 x} d x$ .
[3 marks]
Answer
Total for D4: 8 marks
D5
This question is about $tan x$ and $cot x$ for $0 < x < \frac{π}{2}$ .
(a)
Show that $\int_{0}^{\frac{π}{4}} tan x d x = \frac{1}{2} ln 2$ .
[3 marks]
(b)
Find the exact value of $\int_{\frac{π}{6}}^{\frac{π}{3}} cot x d x$ .
[3 marks]
Answer
(c)
Given that $\int_{0}^{k} tan x d x = ln 2$ , where $0 < k < \frac{π}{2}$ , find the exact value of $k$ .
[3 marks]
Answer
Total for D5: 9 marks
D6
This question is about $\frac{x ^{2}}{( x ^{3} + 1 ) ^{3}}$ .
(a)
Find $\int \frac{x ^{2}}{( x ^{3} + 1 ) ^{3}} d x$ .
[3 marks]
Answer
(b)
Find the value of $\int_{0}^{1} \frac{x ^{2}}{( x ^{3} + 1 ) ^{3}} d x$ .
[2 marks]
Answer
(c)
Show that $\int_{0}^{k} \frac{x ^{2}}{( x ^{3} + 1 ) ^{3}} d x < \frac{1}{6}$ for every $k > 0$ .
[3 marks]
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 $\int_{0}^{\frac{π}{4}} sin^{4} x d x$ .
[9 marks]
Hint 1 · What to try
Write $sin^{4} x = (sin^{2} x)^{2}$ and use $sin^{2} x = \frac{1}{2} - \frac{1}{2} cos 2 x$ .
Hint 2 · The first line
Squaring gives a $cos^{2} 2 x$ term. Use the identity again: $cos^{2} 2 x = \frac{1}{2} + \frac{1}{2} cos 4 x$ .
Hint 3 · The full method
You should reach $sin^{4} x = \frac{3}{8} - \frac{1}{2} cos 2 x + \frac{1}{8} cos 4 x$ . Integrate each term.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the double angle formulae to integrate sin²x, cos²x and sin x cos x.
I can use tan²x = sec²x − 1 and cot²x = cosec²x − 1 before integrating.
I can recognise f′(x)/f(x) and integrate it to ln\|f(x)\|.
I can recognise f′(x)[f(x)]ⁿ, integrate it, and check by differentiating.

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