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.

9 questions · Standard +1.0

4.08c Improper integrals: infinite limits or discontinuous integrands

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OCR FP2 2008 June Q9

12 marks Challenging +1.3

Prove that $\int _ { 0 } ^ { N } \ln ( 1 + x ) \mathrm { d } x = ( N + 1 ) \ln ( N + 1 ) - N$, where $N$ is a positive constant.
\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.
1. By considering the areas of these rectangles, explain why $$\ln 2 + \ln 3 + \ln 4 + \ldots + \ln 70 < \int _ { 0 } ^ { 70 } \ln ( 1 + x ) d x$$
2. By considering the areas of another set of rectangles, show that $$\ln 2 + \ln 3 + \ln 4 + \ldots + \ln 70 > \int _ { 0 } ^ { 69 } \ln ( 1 + x ) d x$$
3. 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

Use integration by parts to show that $\int \ln x \mathrm {~d} x = x \ln x - x + c$, where $c$ is an arbitrary constant.
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

Find $\int x ^ { 3 } \ln x \mathrm {~d} x$.
Explain why $\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x$ is an improper integral.
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

Find $\int x ^ { 2 } \ln x \mathrm {~d} x$.
Explain why $\int _ { 0 } ^ { \mathrm { e } } x ^ { 2 } \ln x \mathrm {~d} x$ is an improper integral.
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

Find $\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$.
Hence evaluate $\int _ { 0 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$, showing the limiting process used.

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

Find $\int _ { 0 } ^ { a } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x$, where $a > 0$.
Write down the value of $\lim _ { a \rightarrow \infty } a ^ { k } \mathrm { e } ^ { - 2 a }$, where $k$ is a positive constant.
Hence find $\int _ { 0 } ^ { \infty } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x$.

AQA FP3 2007 January Q4

8 marks Challenging +1.2

Explain why $\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x$ is an improper integral.
(1 mark)
Use integration by parts to find $\int x ^ { - \frac { 1 } { 2 } } \ln x \mathrm {~d} x$.
(3 marks)
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

Write down the value of $$\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x }$$
Use the substitution $u = x \mathrm { e } ^ { - x } + 1$ to find $\int \frac { \mathrm { e } ^ { - x } ( 1 - x ) } { x \mathrm { e } ^ { - x } + 1 } \mathrm {~d} x$.
Hence evaluate $\int _ { 1 } ^ { \infty } \frac { 1 - x } { x + \mathrm { e } ^ { x } } \mathrm {~d} x$, showing the limiting process used.