Improper integrals with partial fractions (infinite limit)

Use partial fractions to evaluate an improper integral with an infinite limit (typically to infinity), showing the limiting process explicitly. The integrand is a rational function requiring partial fraction decomposition.

5 questions · Standard +0.9

4.08c Improper integrals: infinite limits or discontinuous integrands
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AQA FP3 2011 January Q5
7 marks Standard +0.8
5
  1. Write \(\frac { 4 } { 4 x + 1 } - \frac { 3 } { 3 x + 2 }\) in the form \(\frac { C } { ( 4 x + 1 ) ( 3 x + 2 ) }\), where \(C\) is a constant.
    (l mark)
  2. Evaluate the improper integral $$\int _ { 1 } ^ { \infty } \frac { 10 } { ( 4 x + 1 ) ( 3 x + 2 ) } d x$$ showing the limiting process used and giving your answer in the form \(\ln k\), where \(k\) is a constant.
    (6 marks)
OCR Further Pure Core 1 2022 June Q7
10 marks Challenging +1.2
7
  1. Determine the values of \(A , B , C\) and \(D\) such that \(\frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \equiv \frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { C x + D } { x ^ { 2 } + 9 }\).
  2. In this question you must show detailed reasoning. Hence determine the exact value of \(\int _ { 3 } ^ { \infty } \frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \mathrm { d } x\).
Edexcel CP1 2019 June Q2
7 marks Challenging +1.2
  1. Show that
$$\int _ { 0 } ^ { \infty } \frac { 8 x - 12 } { \left( 2 x ^ { 2 } + 3 \right) ( x + 1 ) } \mathrm { d } x = \ln k$$ where \(k\) is a rational number to be found.
Edexcel CP1 2020 June Q2
7 marks Standard +0.8
  1. (a) Explain why \(\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } d x\) is an improper integral.
    (b) Prove that
$$\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } d x = a \ln b$$ where \(a\) and \(b\) are rational numbers to be determined.
SPS SPS FM Pure 2024 February Q6
7 marks Standard +0.3
  1. Explain why \(\int_1^\infty \frac{1}{x(2x + 5)} dx\) is an improper integral. [1]
  2. Prove that $$\int_1^\infty \frac{1}{x(2x + 5)} dx = a \ln b$$ where \(a\) and \(b\) are rational numbers to be determined. [6]