Improper integral with parts

A question is this type if and only if it asks to evaluate an improper integral (with infinite limit or discontinuity) using integration by parts and showing the limiting process explicitly.

10 questions · Standard +0.9

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OCR FP2 2008 June Q9
12 marks
9
  1. Prove that \(\int _ { 0 } ^ { N } \ln ( 1 + x ) \mathrm { d } x = ( N + 1 ) \ln ( N + 1 ) - N\), where \(N\) is a positive constant.
  2. \includegraphics[max width=\textwidth, alt={}, center]{63a316f6-1c18-4224-930f-0b58112c9f71-4_616_1261_406_482} The diagram shows the curve \(y = \ln ( 1 + x )\), for \(0 \leqslant x \leqslant 70\), together with a set of rectangles of unit width.
    (a) By considering the areas of these rectangles, explain why $$\ln 2 + \ln 3 + \ln 4 + \ldots + \ln 70 < \int _ { 0 } ^ { 70 } \ln ( 1 + x ) d x$$ (b) By considering the areas of another set of rectangles, show that $$\ln 2 + \ln 3 + \ln 4 + \ldots + \ln 70 > \int _ { 0 } ^ { 69 } \ln ( 1 + x ) d x$$ (c) Hence find bounds between which \(\ln ( 70 ! )\) lies. Give the answers correct to 1 decimal place.
AQA FP3 2009 January Q4
6 marks Standard +0.8
4
  1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\), where \(c\) is an arbitrary constant.
  2. Hence evaluate \(\int _ { 0 } ^ { 1 } \ln x \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2008 June Q5
7 marks Standard +0.8
5
  1. Find \(\int x ^ { 3 } \ln x \mathrm {~d} x\).
  2. Explain why \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x\) is an improper integral.
  3. Evaluate \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2011 June Q3
7 marks Standard +0.8
3
  1. Find \(\int x ^ { 2 } \ln x \mathrm {~d} x\).
  2. Explain why \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 2 } \ln x \mathrm {~d} x\) is an improper integral.
  3. Evaluate \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 2 } \ln x \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2012 June Q5
7 marks Standard +0.8
5
  1. Find \(\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x\).
  2. Hence evaluate \(\int _ { 0 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x\), showing the limiting process used.
OCR Further Pure Core 2 Specimen Q5
4 marks Standard +0.8
5 In this question you must show detailed reasoning.
Evaluate \(\int _ { 0 } ^ { \infty } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x\).
[0pt] [You may use the result \(\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x } = 0\).]
OCR Further Pure Core 2 2017 Specimen Q5
4 marks Standard +0.8
5 In this question you must show detailed reasoning. Evaluate \(\int _ { 0 } ^ { \infty } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x\).
[0pt] [You may use the result \(\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x } = 0\).]
AQA FP3 2006 January Q2
8 marks Standard +0.3
2
  1. Find \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), where \(a > 0\).
  2. Write down the value of \(\lim _ { a \rightarrow \infty } a ^ { k } \mathrm { e } ^ { - 2 a }\), where \(k\) is a positive constant.
  3. Hence find \(\int _ { 0 } ^ { \infty } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
AQA FP3 2007 January Q4
8 marks Challenging +1.2
4
  1. Explain why \(\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x\) is an improper integral.
    (1 mark)
  2. Use integration by parts to find \(\int x ^ { - \frac { 1 } { 2 } } \ln x \mathrm {~d} x\).
    (3 marks)
  3. Show that \(\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x\) exists and find its value.
    (4 marks)
AQA FP3 2007 June Q7
7 marks Challenging +1.8
7
  1. Write down the value of $$\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x }$$
  2. Use the substitution \(u = x \mathrm { e } ^ { - x } + 1\) to find \(\int \frac { \mathrm { e } ^ { - x } ( 1 - x ) } { x \mathrm { e } ^ { - x } + 1 } \mathrm {~d} x\).
  3. Hence evaluate \(\int _ { 1 } ^ { \infty } \frac { 1 - x } { x + \mathrm { e } ^ { x } } \mathrm {~d} x\), showing the limiting process used.