A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 11: Integration · Lesson 5

Integration with partial fractions

116 marks

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Quick questions to start.

A1
Needed today
Write $\frac{x + 7}{( x - 1 ) ( x + 3 )}$ in partial fractions.
[3 marks]
Answer
A2
Needed today
Find $\int \frac{3}{x + 2} d x$ .
[2 marks]
Answer
A3
Common slip
Find $\int \frac{4}{2 x - 1} d x$ . Check your answer by differentiating it.
[2 marks]
Answer
A4
Needed today
Write $2 ln 3 - ln 6$ as a single logarithm.
[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 $\int \frac{5 x + 1}{( x - 1 ) ( x + 2 )} d x$ .

Hint Split it first: it is

\frac{2}{x - 1} + \frac{3}{x + 2}

. Each part integrates to a logarithm.

B1
Find $\int \frac{4 x + 5}{( x + 2 ) ( x + 1 )} d x$ .
[4 marks]
Answer

Example 2

Find the exact value of $\int_{2}^{3} \frac{3}{( x - 1 ) ( x + 2 )} d x$ .

Hint It is

\frac{1}{x - 1} - \frac{1}{x + 2}

. Integrate, put in the limits, then use the log laws to make one logarithm.

B2
Find the exact value of $\int_{3}^{4} \frac{5}{( 2 x + 1 ) ( x - 2 )} d x$ . Give your answer as a single logarithm.
[6 marks]
Answer

Example 3

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

Hint Write it as

\frac{A}{x + 1} + \frac{B}{( x + 1 ) ^{2}}

. The second part is a power:

\int (x + 1)^{- 2} d x = - (x + 1)^{- 1}

, not a logarithm.

B3
Find $\int \frac{3 x + 1}{( x + 1 ) ^{2}} d x$ .
[4 marks]
Answer

Example 4

Find $\int \frac{x ^{2}}{x ^{2} - 1} d x$ .

Hint The top has the same degree as the bottom, so divide first:

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

. Then split

\frac{1}{( x - 1 ) ( x + 1 )}

B4
Find the exact value of $\int_{2}^{3} \frac{x ^{2} + 1}{x ^{2} - 1} d x$ .
[6 marks]
Answer

Example 5

Find $\int \frac{1}{x ( x + 1 )} d x$ and write the answer as a single logarithm.

Hint

\frac{1}{x ( x + 1 )} = \frac{1}{x} - \frac{1}{x + 1}

. A difference of logarithms is the logarithm of a quotient.

B5
Find the exact value of $\int_{1}^{4} \frac{2}{x ( x + 2 )} d x$ .
[4 marks]
Answer

Practice

A-Level

C1
Nia writes $\int_{0}^{4} \frac{3}{2 x + 1} d x = [3 ln ∣ 2 x + 1 ∣]_{0}^{4} = 3 ln 9$ . Find her mistake, then find the correct value in the form $k ln 3$ .
[4 marks]
Answer

C2
Max tries to split $\frac{x ^{2} + x}{( x + 2 ) ( x - 1 )}$ as $\frac{A}{x + 2} + \frac{B}{x - 1}$ and cannot make it work. Explain why, then find $\int \frac{x ^{2} + x}{( x + 2 ) ( x - 1 )} d x$ .
[5 marks]
Answer
C3
Find the exact value of $\int_{2}^{3} \frac{1}{x ( x - 1 )} d x$ . Give your answer as a single logarithm.
[4 marks]
Answer
C4
Find $\int \frac{8}{x ^{2} - 16} d x$ , writing your answer as a single logarithm.
[4 marks]
Answer
C5
Find the exact value of $\int_{0}^{1} \frac{x}{( x + 1 ) ^{2}} d x$ .
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
$f (x) = \frac{x + 10}{( 2 x - 1 ) ( x + 3 )}$ .
(a)
Write $f (x)$ in partial fractions.
[3 marks]
Answer
(b)
Find the exact value of $\int_{1}^{2} f (x) d x$ , giving your answer in the form $ln k$ .
[5 marks]
Answer
(c)
Explain why $\int_{0}^{1} f (x) d x$ cannot be found with this method.
[2 marks]
Total for D1: 10 marks

D2
Raj writes $\int_{1}^{3} \frac{4}{( 2 x + 1 ) ( 2 x + 3 )} d x = [2 ln ∣ 2 x + 1 ∣ - 2 ln ∣ 2 x + 3 ∣]_{1}^{3}$ .
Raj is wrong.
(a)
Explain his mistake.
[2 marks]
(b)
Find the exact value of $\int_{1}^{3} \frac{4}{( 2 x + 1 ) ( 2 x + 3 )} d x$ , giving your answer as a single logarithm.
[4 marks]
Answer
(c)
Show that $\int_{1}^{n} \frac{4}{( 2 x + 1 ) ( 2 x + 3 )} d x < ln \frac{5}{3}$ for every $n > 1$ .
[3 marks]
Total for D2: 9 marks
D3
Find the exact value of $\int_{3}^{5} \frac{2 x ^{2} - 3}{x ^{2} - 4} d x$ , giving your answer in the form $a + b ln c$ , where $a$ , $b$ and $c$ are rational.
[7 marks]
Answer
D4
$g (x) = \frac{2 x ^{2} + 3 x + 4}{x ^{2} ( x + 2 )}$ .
(a)
Write $g (x)$ in the form $\frac{A}{x} + \frac{B}{x ^{2}} + \frac{C}{x + 2}$ .
[4 marks]
Answer
(b)
Find the exact value of $\int_{1}^{2} g (x) d x$ , giving your answer in the form $p + q ln 2 + r ln 3$ .
[5 marks]
Answer
Total for D4: 9 marks
D5
This question is about $\int_{0}^{l n 2} \frac{e ^{x}}{( e ^{x} + 1 ) ( e ^{x} + 2 )} d x$ .
(a)
Use the substitution $u = e^{x}$ to show that the integral is $\int_{1}^{2} \frac{1}{( u + 1 ) ( u + 2 )} d u$ .
[2 marks]
(b)
Hence find the exact value of the integral, giving your answer as a single logarithm.
[6 marks]
Answer
Total for D5: 8 marks
D6
$h (x) = \frac{3}{( 1 - x ) ( 1 + 2 x )}$ .
(a)
Write $h (x)$ in partial fractions.
[3 marks]
Answer
(b)
Find the exact value of $\int_{0}^{\frac{1}{2}} h (x) d x$ .
[4 marks]
Answer
(c)
$\int_{0}^{k} h (x) d x$ , with $k > 0$ , has a finite value. State the values of $k$ for which this is true, and give a reason.
[2 marks]
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
$\int_{2}^{k} \frac{4}{x ^{2} - 1} d x = ln 2$ , where $k > 2$ . Find the exact value of $k$ , in the form $\frac{a + b 2}{c}$ .
[9 marks]
Hint 1 · What to try
Split $\frac{4}{x ^{2} - 1}$ into $\frac{2}{x - 1} - \frac{2}{x + 1}$ and integrate.
Hint 2 · The first line
The integral is $2 ln \frac{k - 1}{k + 1} - 2 ln \frac{1}{3} = 2 ln \frac{3 ( k - 1 )}{k + 1}$ . Set this equal to $ln 2$ .
Hint 3 · The full method
So $\frac{3 ( k - 1 )}{k + 1} = 2$ . Solve for $k$ and rationalise the denominator.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can split a fraction into partial fractions and integrate each part to a logarithm.
I can integrate $\frac{k}{a x + b}$ to $\frac{k}{a} ln ∣ a x + b ∣$ without losing the $\frac{1}{a}$ .
I can integrate a repeated factor, where $\frac{1}{( x + a ) ^{2}}$ does not give a logarithm.
I can divide an improper fraction first, and write an exact answer as a single logarithm.

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