Improper integrals with discontinuity

Evaluate an improper integral where the integrand has a discontinuity within or at the boundary of the integration interval (e.g., ln(x) at x=0, 1/√x at x=0), showing the limiting process explicitly.

9 questions · Standard +0.9

4.08c Improper integrals: infinite limits or discontinuous integrands
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AQA FP3 2013 January Q4
7 marks Challenging +1.2
4
  1. Explain why \(\int _ { 0 } ^ { 1 } x ^ { 4 } \ln x \mathrm {~d} x\) is an improper integral.
    (l mark)
  2. Evaluate \(\int _ { 0 } ^ { 1 } x ^ { 4 } \ln x \mathrm {~d} x\), showing the limiting process used.
    (6 marks)
AQA FP3 2014 June Q5
4 marks Challenging +1.3
5
  1. Find \(\int x \cos 8 x \mathrm {~d} x\).
  2. Find \(\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x } \sin 2 x \right]\).
  3. Explain why \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 2 \cot 2 x - \frac { 1 } { x } + x \cos 8 x \right) \mathrm { d } x\) is an improper integral.
  4. Evaluate \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 2 \cot 2 x - \frac { 1 } { x } + x \cos 8 x \right) \mathrm { d } x\), showing the limiting process used. Give your answer as a single term.
    [0pt] [4 marks]
AQA FP1 2006 January Q2
7 marks Standard +0.8
2
  1. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
    1. \(\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\);
    2. \(\int _ { 0 } ^ { 9 } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\).
  2. Explain briefly why the integrals in part (a) are improper integrals.
AQA FP1 2007 June Q8
8 marks Challenging +1.2
8 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
  1. \(\quad \int _ { 0 } ^ { 1 } \left( x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } x\);
  2. \(\int _ { 0 } ^ { 1 } \frac { x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } } { x } \mathrm {~d} x\).
AQA FP1 2015 June Q2
5 marks Challenging +1.2
2
  1. Explain why \(\int _ { 0 } ^ { 4 } \frac { x - 4 } { x ^ { 1.5 } } \mathrm {~d} x\) is an improper integral.
  2. Either find the value of the integral \(\int _ { 0 } ^ { 4 } \frac { x - 4 } { x ^ { 1.5 } } \mathrm {~d} x\) or explain why it does not have a finite value.
    [0pt] [4 marks]
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OCR MEI Further Pure Core 2024 June Q7
5 marks Challenging +1.2
7
  1. Explain why \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt [ 3 ] { x - 2 } } \mathrm {~d} x\) is an improper integral.
  2. In this question you must show detailed reasoning. Use an appropriate limit argument to evaluate this integral.
AQA Further Paper 1 2021 June Q10
6 marks Challenging +1.2
Evaluate the improper integral $$\int_0^8 \ln x \, \mathrm{d}x$$ showing the limiting process. [6 marks]
SPS SPS FM Pure 2021 May Q3
5 marks Moderate -0.3
In this question you must show detailed reasoning. Show that $$\int_5^{\infty} (x - 1)^{-\frac{3}{2}} dx = 1$$ [5]
SPS SPS FM Pure 2022 June Q1
7 marks Standard +0.3
  1. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
    1. \(\int_0^9 \frac{1}{\sqrt{x}} dx\); [3 marks]
    2. \(\int_0^9 \frac{1}{x\sqrt{x}} dx\). [3 marks]
  2. Explain briefly why the integrals in part (a) are improper integrals. [1 mark]