Partial fractions with parameter

Express a rational function containing a parameter (constant a or k) in partial fractions, then integrate to show a result independent of or in terms of the parameter.

1 questions · Challenging +1.2

1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions
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CAIE P3 2024 March Q10
10 marks Challenging +1.2
10 Let \(f ( x ) = \frac { 36 a ^ { 2 } } { ( 2 a + x ) ( 2 a - x ) ( 5 a - 2 x ) }\), where \(a\) is a positive constant.
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find the exact value of \(\int _ { - a } ^ { a } f ( x ) d x\), giving your answer in the form plnq+rlns where \(p\) and \(r\) are integers and \(q\) and \(s\) are prime numbers.