Partial fractions with linear factors – decompose and integrate (indefinite)

Express a proper rational function with distinct or repeated linear factors in partial fractions, then find an indefinite integral.

6 questions · Moderate -0.6

1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions
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OCR C4 2008 January Q2
5 marks Moderate -0.8
2
  1. Express \(\frac { x } { ( x + 1 ) ( x + 2 ) }\) in partial fractions.
  2. Hence find \(\int \frac { x } { ( x + 1 ) ( x + 2 ) } \mathrm { d } x\).
AQA C4 2005 June Q2
6 marks Moderate -0.8
2
  1. Express \(\frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) }\) in the form \(\frac { A } { x + 3 } + \frac { B } { 2 x - 1 }\).
  2. Hence find \(\int \frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) } \mathrm { d } x\).
OCR MEI Paper 3 2019 June Q3
6 marks Moderate -0.8
3
  1. Express \(\frac { 1 } { ( x + 2 ) ( x + 3 ) }\) in partial fractions.
  2. Find \(\int \frac { 1 } { ( x + 2 ) ( x + 3 ) } \mathrm { d } x\) in the form \(\ln ( \mathrm { f } ( x ) ) + c\), where \(c\) is the constant of integration and \(\mathrm { f } ( x )\) is a function to be determined.
Edexcel C4 2013 June Q6
11 marks Moderate -0.3
    1. Express \(\frac{7x}{(x + 3)(2x - 1)}\) in partial fractions. [3]
    2. Given that \(x > \frac{1}{2}\), find $$\int \frac{7x}{(x + 3)(2x - 1)} \, dx$$ [3]
  2. Using the substitution \(u^3 = x\), or otherwise, find $$\int \frac{1}{x + x^3} \, dx, \quad x > 0$$ [5]
AQA C4 2010 June Q3
8 marks Moderate -0.3
    1. Express \(\frac{7x - 3}{(x + 1)(3x - 2)}\) in the form \(\frac{A}{x + 1} + \frac{B}{3x - 2}\). [3 marks]
    2. Hence find \(\int \frac{7x - 3}{(x + 1)(3x - 2)} dx\). [2 marks]
  2. Express \(\frac{6x^2 + x + 2}{2x^2 - x + 1}\) in the form \(P + \frac{Qx + R}{2x^2 - x + 1}\). [3 marks]
SPS SPS SM Pure 2020 October Q6
5 marks Moderate -0.8
  1. Express \(\frac{x}{(x + 1)(x + 2)}\) in partial fractions. [3]
  2. Hence find \(\int \frac{x}{(x + 1)(x + 2)} dx\). [2]