Partial fractions to find specific parameter

Use partial fractions and integration to find the value of an unknown parameter k or a that satisfies a given condition on a definite integral.

2 questions · Challenging +1.0

1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions
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Edexcel P4 2023 June Q3
11 marks Standard +0.8
3. $$\mathrm { f } ( x ) = \frac { 8 x - 5 } { ( 2 x - 1 ) ( 4 x - 3 ) } \quad x > 1$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\)
  3. Use the answer to part (b) to find the value of \(k\) for which $$\int _ { k } ^ { 3 k } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 2 } \ln 20$$
Edexcel Paper 2 2023 June Q10
7 marks Challenging +1.2
  1. \(\mathrm { f } ( x ) = \frac { 3 k x - 18 } { ( x + 4 ) ( x - 2 ) } \quad\) where \(k\) is a positive constant
    1. Express \(\mathrm { f } ( x )\) in partial fractions in terms of \(k\).
    2. Hence find the exact value of \(k\) for which
    $$\int _ { - 3 } ^ { 1 } f ( x ) d x = 21$$