Write down the integral: (5)/(x - 2) dx Integrate using the standard logarithm result: (5)/(x - 2) dx=5 (x - 2 ) Add the constant of integration: 5 (x - 2 ) + c Answer: 5 (x - 2 ) + c

Write down the integral: (3)/(x + 1) dx Integrate using the standard logarithm result: (3)/(x + 1) dx=3 (x + 1 ) Add the constant of integration: 3 (x + 1 ) + c Answer: 3 (x + 1 ) + c

Write down the integral: (7)/(x - 4) dx Integrate using the standard logarithm result: (7)/(x - 4) dx=7 (x - 4 ) Add the constant of integration: 7 (x - 4 ) + c Answer: 7 (x - 4 ) + c

Find (3 x + 1)/((x - 1) (x + 2)) dx.

Write the partial fraction decomposition: (3 x + 1)/((x - 1) (x + 2)) (5)/(3 (x + 2)) + (4)/(3 (x - 1)) Integrate each term: (3 x + 1)/((x - 1) (x + 2)) dx=4 (x - 1 )3 + 5 (x + 2 )3 Add the constant of integration: 4 (x - 1 )3 + 5 (x + 2 )3 + c Answer: 4 (x - 1 )3 + 5 (x + 2 )3 + c

Find (5)/((x - 2) (x - 1)) dx.

Write the partial fraction decomposition: (5)/((x - 2) (x - 1)) - (5)/(x - 1) + (5)/(x - 2) Integrate each term: (5)/((x - 2) (x - 1)) dx=5 (x - 2 ) - 5 (x - 1 ) Add the constant of integration: 5 (x - 2 ) - 5 (x - 1 ) + c Answer: 5 (x - 2 ) - 5 (x - 1 ) + c

Integration with partial fractions Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Integration with partial fractions questions. See exactly how to solve problems on partial fractions, integration.

partial fractionsintegration

A-Level70 questionsStep-by-step solutions

Question 1

2 markseasy

Find

\int \frac{5}{x - 2}\,dx

Worked solution

Write down the integral
$\int \frac{5}{x - 2}\,dx$
We integrate this rational function by first splitting it into partial fractions.
Integrate using the standard logarithm result
$\int \frac{5}{x - 2}\,dx=5 \ln{\left(x - 2 \right)}$
Recall \int \frac{k}{x-a}\,dx=k\ln|x-a|+c.
Add the constant of integration
$5 \ln{\left(x - 2 \right)} + c$
This is an indefinite integral, so we include +c.

Answer

5 \ln{\left(x - 2 \right)} + c

Question 2

2 markseasy

Find

\int \frac{3}{x + 1}\,dx

Worked solution

Write down the integral
$\int \frac{3}{x + 1}\,dx$
We integrate this rational function by first splitting it into partial fractions.
Integrate using the standard logarithm result
$\int \frac{3}{x + 1}\,dx=3 \ln{\left(x + 1 \right)}$
Recall \int \frac{k}{x-a}\,dx=k\ln|x-a|+c.
Add the constant of integration
$3 \ln{\left(x + 1 \right)} + c$
This is an indefinite integral, so we include +c.

Answer

3 \ln{\left(x + 1 \right)} + c

Question 3

2 markseasy

Find

\int \frac{7}{x - 4}\,dx

Worked solution

Write down the integral
$\int \frac{7}{x - 4}\,dx$
We integrate this rational function by first splitting it into partial fractions.
Integrate using the standard logarithm result
$\int \frac{7}{x - 4}\,dx=7 \ln{\left(x - 4 \right)}$
Recall \int \frac{k}{x-a}\,dx=k\ln|x-a|+c.
Add the constant of integration
$7 \ln{\left(x - 4 \right)} + c$
This is an indefinite integral, so we include +c.

Answer

7 \ln{\left(x - 4 \right)} + c

Question 4

2 markseasy

Find

\int \frac{3 x + 1}{\left(x - 1\right) \left(x + 2\right)}\,dx

Worked solution

Write the partial fraction decomposition
$\frac{3 x + 1}{\left(x - 1\right) \left(x + 2\right)}\equiv \frac{5}{3 \left(x + 2\right)} + \frac{4}{3 \left(x - 1\right)}$
Each simple fraction is now easy to integrate.
Integrate each term
$\int \frac{3 x + 1}{\left(x - 1\right) \left(x + 2\right)}\,dx=\frac{4 \ln{\left(x - 1 \right)}}{3} + \frac{5 \ln{\left(x + 2 \right)}}{3}$
Each simple fraction integrates to a logarithm.
Add the constant of integration
$\frac{4 \ln{\left(x - 1 \right)}}{3} + \frac{5 \ln{\left(x + 2 \right)}}{3} + c$
This is an indefinite integral, so we include +c.

Answer

\frac{4 \ln{\left(x - 1 \right)}}{3} + \frac{5 \ln{\left(x + 2 \right)}}{3} + c

Question 5

2 markseasy

Find

\int \frac{5}{\left(x - 2\right) \left(x - 1\right)}\,dx

Worked solution

Write the partial fraction decomposition
$\frac{5}{\left(x - 2\right) \left(x - 1\right)}\equiv - \frac{5}{x - 1} + \frac{5}{x - 2}$
Each simple fraction is now easy to integrate.
Integrate each term
$\int \frac{5}{\left(x - 2\right) \left(x - 1\right)}\,dx=5 \ln{\left(x - 2 \right)} - 5 \ln{\left(x - 1 \right)}$
Each simple fraction integrates to a logarithm.
Add the constant of integration
$5 \ln{\left(x - 2 \right)} - 5 \ln{\left(x - 1 \right)} + c$
This is an indefinite integral, so we include +c.

Answer

5 \ln{\left(x - 2 \right)} - 5 \ln{\left(x - 1 \right)} + c

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