Repeated linear factor with distinct linear factor – decompose and integrate

Denominator contains one repeated linear factor (x-a)² and one or more distinct linear factors; express in partial fractions of the form A/(x-b) + B/(x-a) + C/(x-a)², then evaluate a definite or indefinite integral.

16 questions · Standard +0.2

1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions
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Edexcel C4 2013 June Q1
4 marks Moderate -0.3
  1. Express in partial fractions
$$\frac { 5 x + 3 } { ( 2 x + 1 ) ( x + 1 ) ^ { 2 } }$$
OCR C4 Q5
10 marks Standard +0.3
5. $$f ( x ) = \frac { 7 + 3 x + 2 x ^ { 2 } } { ( 1 - 2 x ) ( 1 + x ) ^ { 2 } } , \quad | x | > \frac { 1 } { 2 }$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that $$\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = p - \ln q$$ where \(p\) is rational and \(q\) is an integer.
OCR C4 2009 June Q6
9 marks Moderate -0.3
6 The expression \(\frac { 4 x } { ( x - 5 ) ( x - 3 ) ^ { 2 } }\) is denoted by \(\mathrm { f } ( x )\).
  1. Express f \(( x )\) in the form \(\frac { A } { x - 5 } + \frac { B } { x - 3 } + \frac { C } { ( x - 3 ) ^ { 2 } }\), where \(A , B\) and \(C\) are constants.
  2. Hence find the exact value of \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).
OCR C4 2012 June Q9
9 marks Standard +0.3
9
  1. Express \(\frac { x ^ { 2 } - x - 11 } { ( x + 1 ) ( x - 2 ) ^ { 2 } }\) in partial fractions.
  2. Find the exact value of \(\int _ { 3 } ^ { 4 } \frac { x ^ { 2 } - x - 11 } { ( x + 1 ) ( x - 2 ) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are rational numbers.
OCR C4 2014 June Q9
9 marks Standard +0.3
9 Express \(\frac { 2 + x ^ { 2 } } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }\) in partial fractions and hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } } \frac { 2 + x ^ { 2 } } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \frac { 3 } { 2 } + \frac { 1 } { 3 }\).
Edexcel C4 Q6
11 marks Standard +0.3
6. $$f ( x ) = \frac { 15 - 17 x } { ( 2 + x ) ( 1 - 3 x ) ^ { 2 } } , \quad x \neq - 2 , \quad x \neq \frac { 1 } { 3 }$$
  1. Find the values of the constants \(A , B\) and \(C\) such that $$\mathrm { f } ( x ) = \frac { A } { 2 + x } + \frac { B } { 1 - 3 x } + \frac { C } { ( 1 - 3 x ) ^ { 2 } }$$
  2. Find the value of $$\int _ { - 1 } ^ { 0 } f ( x ) d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.
    6. continued
Edexcel C4 Q3
11 marks Standard +0.3
3. $$f ( x ) = \frac { 7 + 3 x + 2 x ^ { 2 } } { ( 1 - 2 x ) ( 1 + x ) ^ { 2 } } , \quad | x | > \frac { 1 } { 2 }$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that $$\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = p - \ln q$$ where \(p\) is rational and \(q\) is an integer.
    3. continued
AQA C4 2009 January Q3
13 marks Standard +0.3
3
    1. Express \(\frac { 2 x + 7 } { x + 2 }\) in the form \(A + \frac { B } { x + 2 }\), where \(A\) and \(B\) are integers. (2 marks)
    2. Hence find \(\int \frac { 2 x + 7 } { x + 2 } \mathrm {~d} x\).
    1. Express \(\frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } }\) in the form \(\frac { P } { 1 + 3 x } + \frac { Q } { 5 - x } + \frac { R } { ( 5 - x ) ^ { 2 } }\), where \(P , Q\) and \(R\) are constants.
    2. Hence find \(\int \frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } } \mathrm {~d} x\).
WJEC Unit 3 2019 June Q1
Moderate -0.3
a) Express \(\frac { 9 } { ( x - 1 ) ( x + 2 ) ^ { 2 } }\) in terms of partial fractions. b) Find \(\int \frac { 9 } { ( x - 1 ) ( x + 2 ) ^ { 2 } } \mathrm {~d} x\).
Edexcel C4 Q7
16 marks Standard +0.8
$$\text{f}(x) = \frac{25}{(3 + 2x)^2(1 - x)}, \quad |x| < 1.$$
  1. Express f(x) as a sum of partial fractions. [4]
  2. Hence find \(\int \text{f}(x) \, dx\). [5]
  3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^2\). Give each coefficient as a simplified fraction. [7]
AQA Paper 3 2019 June Q7
8 marks Standard +0.3
  1. Express \(\frac{4x + 3}{(x - 1)^2}\) in the form \(\frac{A}{x - 1} + \frac{B}{(x - 1)^2}\) [3 marks]
  2. Show that $$\int_3^4 \frac{4x + 3}{(x - 1)^2} \, dx = p + \ln q$$ where \(p\) and \(q\) are rational numbers. [5 marks]
WJEC Unit 3 2018 June Q5
8 marks Moderate -0.3
  1. Show that $$\frac{3x}{(x-1)(x-4)^2} = \frac{A}{(x-1)} + \frac{B}{(x-4)} + \frac{C}{(x-4)^2},$$ where \(A\), \(B\) and \(C\) are constants to be found. [3]
  2. Evaluate \(\int_5^7 \frac{3x}{(x-1)(x-4)^2} \, dx\), giving your answer correct to 3 decimal places. [5]
SPS SPS SM 2021 November Q2
6 marks Moderate -0.3
  1. Express \(\frac{5x+7}{(x+3)(x+1)^2}\) in partial fractions. In this question you must show all of your algebraic steps clearly. [3] The function \(f(x) = \frac{2-6x+5x^2}{x^2(1-2x)}\) can be written in the form; $$f(x) = \frac{-2}{x} + \frac{2}{x^2} + \frac{1}{1-2x}$$
  2. Hence find the exact value of \(\int_2^3 \frac{2-6x+5x^2}{x^2(1-2x)} dx\) [3]
SPS SPS FM 2023 January Q3
5 marks Standard +0.8
Express \(\frac{x^2}{(x-1)^2(x-2)}\) in partial fractions. [5]
SPS SPS FM 2023 February Q3
5 marks Standard +0.3
Express \(\frac{(x-7)(x-2)}{(x+2)(x-1)^2}\) in partial fractions. [5]
SPS SPS SM Pure 2023 June Q3
3 marks Standard +0.3
Express in partial fractions, $$\frac{9x^2}{(x-1)^2(2x+1)}$$ [3]