Partial fractions with repeated linear factor

Decompose a rational function that has a repeated linear factor in the denominator (e.g., (x+1)²) into partial fractions, then integrate.

16 questions · Standard +0.3

1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions
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CAIE P3 2002 June Q6
10 marks Standard +0.3
6 Let \(\mathrm { f } ( x ) = \frac { 4 x } { ( 3 x + 1 ) ( x + 1 ) ^ { 2 } }\).
  1. Express \(f ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = 1 - \ln 2\).
CAIE P3 2015 June Q10
10 marks Standard +0.3
10 Let \(\mathrm { f } ( x ) = \frac { 11 x + 7 } { ( 2 x - 1 ) ( x + 2 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 4 } + \ln \left( \frac { 9 } { 4 } \right)\).
CAIE P3 2019 June Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 10 x + 9 } { ( 2 x + 1 ) ( 2 x + 3 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 2 } \ln \frac { 9 } { 5 } + \frac { 1 } { 5 }\).
CAIE P3 2006 November Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 7 x + 4 } { ( 2 x + 1 ) ( x + 1 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 2 + \ln \frac { 5 } { 3 }\).
CAIE P3 2018 November Q9
10 marks Standard +0.3
9 Let \(\mathrm { f } ( x ) = \frac { 6 x ^ { 2 } + 8 x + 9 } { ( 2 - x ) ( 3 + 2 x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence, showing all necessary working, show that \(\int _ { - 1 } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x = 1 + \frac { 1 } { 2 } \ln \left( \frac { 3 } { 4 } \right)\).
CAIE P3 2023 June Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 3 - 3 x ^ { 2 } } { ( 2 x + 1 ) ( x + 2 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find the exact value of \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.
CAIE P3 2023 June Q9
10 marks Standard +0.3
9 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + 17 x - 17 } { ( 1 + 2 x ) ( 2 - x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 5 } { 2 } - \ln 72\). \includegraphics[max width=\textwidth, alt={}, center]{60bb482b-fa41-42ea-a112-62851e5a19aa-16_524_725_269_696} The diagram shows the curve \(y = ( x + 5 ) \sqrt { 3 - 2 x }\) and its maximum point \(M\).
Edexcel C34 2018 October Q5
10 marks Standard +0.3
5. $$f ( x ) = \frac { 4 x ^ { 2 } + 5 x + 3 } { ( x + 2 ) ( 1 - x ) ^ { 2 } } \equiv \frac { A } { ( x + 2 ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }$$
  1. Find the values of the constants \(A\), \(B\) and \(C\).
    1. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
    2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\), writing your answer in the form \(p + \ln q\), where \(p\) and \(q\) are constants.
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Edexcel P4 2022 January Q4
9 marks Standard +0.3
4. $$\mathrm { f } ( x ) = \frac { 4 - 4 x } { x ( x - 2 ) ^ { 2 } } \quad x > 2$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\)
  3. Find $$\int _ { 3 } ^ { 5 } f ( x ) d x$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are rational numbers to be found.
Edexcel P4 2024 January Q2
10 marks Standard +0.3
  1. Given that
$$\frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \equiv \frac { A } { x - 2 } + \frac { B } { 2 x + 1 } + \frac { C } { ( 2 x + 1 ) ^ { 2 } }$$
  1. find the values of the constants \(A , B\) and \(C\).
  2. Hence find the exact value of $$\int _ { 7 } ^ { 12 } \frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \mathrm {~d} x$$ giving your answer in the form \(p \ln q + r\) where \(p\), \(q\) and \(r\) are rational numbers.
Edexcel C4 2012 June Q1
10 marks Standard +0.3
1. $$\mathrm { f } ( x ) = \frac { 1 } { x ( 3 x - 1 ) ^ { 2 } } = \frac { A } { x } + \frac { B } { ( 3 x - 1 ) } + \frac { C } { ( 3 x - 1 ) ^ { 2 } }$$
  1. Find the values of the constants \(A , B\) and \(C\).
    1. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
    2. Find \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\), leaving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants. 1 \(f ( x ) = \frac { 1 } { x ( 3 x - 1 ) ^ { 2 } } = \frac { A } { x } + \frac { } { ( 3 x }\)
      1. Find the values of the constants \(A , B\) and \(C\).
Edexcel C4 Specimen Q6
11 marks Standard +0.3
6. Given that $$\frac { 11 x - 1 } { ( 1 - x ) ^ { 2 } ( 2 + 3 x ) } \equiv \frac { A } { ( 1 - x ) ^ { 2 } } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 2 + 3 x ) }$$
  1. find the values of \(A , B\) and \(C\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 11 x - 1 } { ( 1 - x ) ^ { 2 } ( 2 + 3 x ) } \mathrm { d } x\), giving your answer in the form \(k + \ln a\), where \(k\) is an integer and \(a\) is a simplified fraction.
Edexcel P4 2018 Specimen Q3
10 marks Standard +0.3
3. $$\mathrm { f } ( x ) = \frac { 1 } { x ( 3 x - 1 ) ^ { 2 } } = \frac { A } { x } + \frac { B } { ( 3 x - 1 ) } + \frac { C } { ( 3 x - 1 ) ^ { 2 } }$$
  1. Find the values of the constants \(A , B\) and \(C\)
    1. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\)
    2. Find \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants.
      (6)
CAIE P3 2018 November Q9
10 marks Standard +0.3
Let \(f(x) = \frac{6x^2 + 8x + 9}{(2 - x)(3 + 2x)^2}\).
  1. Express \(f(x)\) in partial fractions. [5]
  2. Hence, showing all necessary working, show that \(\int_{-1}^0 f(x) dx = 1 + \frac{1}{2}\ln\left(\frac{4}{3}\right)\). [5]
OCR C4 2007 January Q6
7 marks Moderate -0.3
  1. Express \(\frac{2x + 1}{(x - 3)^2}\) in the form \(\frac{A}{x - 3} + \frac{B}{(x - 3)^2}\), where \(A\) and \(B\) are constants. [3]
  2. Hence find the exact value of \(\int_4^{10} \frac{2x + 1}{(x - 3)^2} \, dx\), giving your answer in the form \(a + b \ln c\), where \(a\), \(b\) and \(c\) are integers. [4]
OCR C4 Q5
10 marks Standard +0.3
$$f(x) = \frac{15-17x}{(2+x)(1-3x)^2}, \quad x \neq -2, \quad x \neq \frac{1}{3}.$$
  1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$f(x) = \frac{A}{2+x} + \frac{B}{1-3x} + \frac{C}{(1-3x)^2}.$$ [5]
  2. Find the value of $$\int_{-1}^{0} f(x) \, dx,$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers. [5]