Improper integrals with infinite upper limit (power/logarithm functions)

Evaluate an improper integral with an infinite upper limit where the integrand involves power functions or logarithms (e.g., x^(-n), ln(x)/x^n), determining convergence and finding the value using limiting arguments.

6 questions · Standard +0.5

4.08c Improper integrals: infinite limits or discontinuous integrands
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AQA FP3 2010 January Q6
9 marks Challenging +1.2
6
  1. Explain why \(\int _ { 1 } ^ { \infty } \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x\) is an improper integral.
    1. Show that the substitution \(y = \frac { 1 } { x }\) transforms \(\int \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x\) into \(\int 2 y \ln y \mathrm {~d} y\).
    2. Evaluate \(\int _ { 0 } ^ { 1 } 2 y \ln y \mathrm {~d} y\), showing the limiting process used.
    3. Hence write down the value of \(\int _ { 1 } ^ { \infty } \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x\).
AQA FP3 2016 June Q6
7 marks Standard +0.8
6
  1. Use the substitution \(a = \frac { 1 } { p }\) to find \(\lim _ { p \rightarrow \infty } \left[ \frac { \ln p } { p ^ { k } } \right]\), where \(k > 0\).
  2. Evaluate the improper integral \(\int _ { 1 } ^ { \infty } \frac { \ln x } { x ^ { 7 } } \mathrm {~d} x\), showing the limiting process used.
    [0pt] [4 marks]
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AQA FP1 2011 January Q2
6 marks Standard +0.3
2
  1. Find, in terms of \(p\) and \(q\), the value of the integral \(\int _ { p } ^ { q } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\).
  2. Show that only one of the following improper integrals has a finite value, and find that value:
    1. \(\int _ { 0 } ^ { 2 } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\);
    2. \(\int _ { 2 } ^ { \infty } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\).
AQA FP1 2012 January Q2
5 marks Standard +0.3
2 Show that only one of the following improper integrals has a finite value, and find that value:
  1. \(\quad \int _ { 8 } ^ { \infty } x ^ { - \frac { 2 } { 3 } } \mathrm {~d} x\);
  2. \(\quad \int _ { 8 } ^ { \infty } x ^ { - \frac { 4 } { 3 } } \mathrm {~d} x\).
AQA FP1 2013 January Q4
4 marks Standard +0.3
4 Show that the improper integral \(\int _ { 25 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\) has a finite value and find that value.
AQA FP1 2008 June Q3
7 marks Standard +0.3
3 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
  1. \(\int _ { 9 } ^ { \infty } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\);
  2. \(\int _ { 9 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\).