Repeated linear factor only (two-term denominator)

Denominator is a single repeated linear factor (x-a)² with no other distinct linear factor, requiring the form A/(x-a) + B/(x-a)², then integrate.

2 questions · Moderate -0.3

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OCR C4 2011 January Q2
7 marks Moderate -0.3
2
  1. Express \(\frac { 7 - 2 x } { ( x - 2 ) ^ { 2 } }\) in the form \(\frac { A } { x - 2 } + \frac { B } { ( x - 2 ) ^ { 2 } }\), where \(A\) and \(B\) are constants.
  2. Hence find the exact value of \(\int _ { 4 } ^ { 5 } \frac { 7 - 2 x } { ( x - 2 ) ^ { 2 } } \mathrm {~d} x\).
Pre-U Pre-U 9794/2 2012 Specimen Q8
9 marks Moderate -0.3
8
  1. Express \(\frac { 3 x + 2 } { ( x - 2 ) ^ { 2 } }\) in the form \(\frac { A } { x - 2 } + \frac { B } { ( x - 2 ) ^ { 2 } }\) where \(A\) and \(B\) are constants.
  2. Hence find the exact value of \(\int _ { 6 } ^ { 10 } \frac { 3 x + 2 } { ( x - 2 ) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.