A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 11: Integration · Lesson 1

Integrating standard functions and f(ax + b)

114 marks

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Quick questions to start.

A1
Needed today
Find $\frac{d}{d x} (tan 4 x)$ .
[2 marks]
Answer
A2
Needed today
Find $\int (6 x^{2} - \frac{4}{x ^{3}}) d x$ .
[2 marks]
Answer
A3
Common slip
Differentiate $ln 5 x$ .
[2 marks]
Answer
A4
Needed today
Write $\frac{3}{x}$ as a power of $x$ , then find $\int \frac{3}{x} d x$ .
[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 $\int (3 e^{x} - \frac{2}{x} + cos x) d x$ .

Hint Each term is a standard result:

\int e^{x} d x = e^{x}

\int \frac{1}{x} d x = ln ∣ x ∣

\int cos x d x = sin x

B1
Find $\int (5 sec^{2} x + \frac{3}{x} - 2 sin x) d x$ .
[4 marks]
Answer

Example 2

Find $\int (sec x tan x - 4 cosec x cot x) d x$ .

Hint Read the derivatives backwards. The derivative of

sec x

sec x tan x

, and the derivative of

cosec x

- cosec x cot x

B2
Find $\int (2 cosec^{2} x - 3 sec x tan x) d x$ .
[4 marks]
Answer

Example 3

Find $\int (e^{3 x + 2} + cos (4 x - 1)) d x$ .

Hint Integrate as if the bracket were

x

, then divide by the number in front of

x

. Check by differentiating: the chain rule multiplies by that number.

B3
Find $\int (sin (5 x + 2) + e^{1 - 2 x}) d x$ .
[5 marks]
Answer

Example 4

Find $\int \frac{4}{2 x + 3} d x$ .

Hint

\int \frac{1}{a x + b} d x = \frac{1}{a} ln ∣ a x + b ∣

. Do not forget to divide by

a

B4
Find $\int (\frac{6}{3 x - 1} + \frac{1}{( 2 x + 1 ) ^{3}}) d x$ .
[5 marks]
Answer

Example 5

Find the exact value of $\int_{0}^{\frac{π}{6}} sec^{2} 2 x d x$ .

Hint Integrate, then put in the limits. Use exact values:

tan \frac{π}{3} = 3

B5
Find the exact value of $\int_{1}^{4} \frac{3}{2 x + 1} d x$ , giving your answer in the form $k ln 3$ .
[5 marks]
Answer

Practice

A-Level

C1
Jo writes: " $\int e^{4 x} d x = 4 e^{4 x} + c$ ." Explain her mistake. Then find the exact value of $\int_{0}^{\frac{1}{2}} e^{4 x} d x$ .
[4 marks]
Answer

C2
Sam writes: " $\int \frac{1}{3 x + 2} d x = ln ∣3 x + 2∣ + c$ ." Show, by differentiating his answer, that he is wrong. Then find the exact value of $\int_{0}^{2} \frac{1}{3 x + 2} d x$ in the form $k ln 2$ .
[4 marks]
Answer
C3
Find the value of $\int_{0}^{1} (2 x + 1)^{4} d x$ . A student gets $48.4$ . Say what the student has done.
[4 marks]
Answer
C4
Find $\int \frac{( x + 2 ) ^{2}}{x} d x$ , $x > 0$ .
[4 marks]
Answer
C5
Given that $a > 1$ and $\int_{1}^{a} \frac{4}{2 x - 1} d x = ln 25$ , find the value of $a$ .
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The curve $C$ has equation $y = f (x)$ , $x > 0$ , where $f^{'} (x) = 6 e^{3 x} - \frac{4}{x}$ . The point $(\frac{1}{3}, 2 e)$ lies on $C$ .
(a)
Show that $f (x) = 2 e^{3 x} - 4 ln 3 x$ .
[5 marks]
(b)
Find the exact value of $f (1)$ .
[2 marks]
Answer
(c)
Hence find the exact value of $\int_{\frac{1}{3}}^{1} f^{'} (x) d x$ .
[2 marks]
Answer
Total for D1: 9 marks

D2
Lily writes: " $\int sin 3 x d x = 3 cos 3 x + c$ , so $\int_{0}^{\frac{π}{6}} sin 3 x d x = 3 cos \frac{π}{2} - 3 cos 0 = - 3$ ."
Lily is wrong.
(a)
Explain the two mistakes Lily has made.
[2 marks]
(b)
Find the correct value of $\int_{0}^{\frac{π}{6}} sin 3 x d x$ .
[3 marks]
Answer
(c)
Hence find the exact value of $\int_{0}^{\frac{π}{6}} (2 sin 3 x + 3 sec^{2} 2 x) d x$ .
[4 marks]
Answer
Total for D2: 9 marks
D3
Find the exact value of $\int_{1}^{2} (\frac{4}{2 x - 1} + e^{2 - x}) d x$ . Give your answer in the form $a ln 3 + e + b$ , where $a$ and $b$ are integers.
[7 marks]
Answer
D4
This question is about $\int_{0}^{k} \frac{a}{( 3 x + 1 ) ^{2}} d x$ , where $a$ and $k$ are positive constants.
(a)
Show that $\int_{0}^{1} \frac{a}{( 3 x + 1 ) ^{2}} d x = \frac{a}{4}$ .
[4 marks]
(b)
Given that $a = 12$ and $\int_{0}^{k} \frac{12}{( 3 x + 1 ) ^{2}} d x = 3$ , find $k$ .
[3 marks]
Answer
(c)
Explain why $\int_{0}^{k} \frac{12}{( 3 x + 1 ) ^{2}} d x < 4$ for every $k > 0$ .
[2 marks]
Total for D4: 9 marks
D5
This question is about $cosec 2 x$ .
(a)
Show that $\frac{d}{d x} (cosec 2 x) = - 2 cosec 2 x cot 2 x$ .
[2 marks]
(b)
Hence find $\int cosec 2 x cot 2 x d x$ .
[2 marks]
Answer
(c)
Find the exact value of $\int_{\frac{π}{12}}^{\frac{π}{4}} (cosec 2 x cot 2 x + cosec^{2} 2 x) d x$ .
[4 marks]
Answer
Total for D5: 8 marks
D6
This question is about $(e^{x} + e^{- x})^{2}$ .
(a)
Find $\int (e^{x} + e^{- x})^{2} d x$ .
[3 marks]
Answer
(b)
Hence show that $\int_{0}^{l n 2} (e^{x} + e^{- x})^{2} d x = \frac{15}{8} + 2 ln 2$ .
[4 marks]
(c)
Given that $\int_{0}^{k} e^{2 x} d x = 4$ , find the exact value of $k$ .
[3 marks]
Answer
Total for D6: 10 marks

Extension

Stretch

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

E1
For each integer $n \geq 0$ , let $I_{n} = \int_{n}^{n + 1} \frac{4}{2 x + 1} d x$ . Find the least integer $n$ for which $I_{n} < 0.1$ . Then find, in the form $a ln 3$ , the exact value of $I_{0} + I_{1} + I_{2} + \dots + I_{39}$ .
[9 marks]
Hint 1 · What to try
$I_{n} = [2 ln (2 x + 1)]_{n}^{n + 1} = 2 ln (\frac{2 n + 3}{2 n + 1})$ .
Hint 2 · The first line
$I_{n} < 0.1$ means $\frac{2 n + 3}{2 n + 1} < e^{0.05}$ , which is $1 + \frac{2}{2 n + 1} < e^{0.05}$ . Solve for $n$ and test the integers either side.
Hint 3 · The full method
The sum of the integrals over $[0, 1], [1, 2], \dots, [39, 40]$ is one integral from $0$ to $40$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can integrate eˣ, 1/x and the six standard trigonometric results.
I can integrate f(ax + b) by integrating f and dividing by a.
I can write ln\|x\| with the modulus and use log laws to give an exact answer.
I can find an unknown limit or constant from the value of a definite integral.

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