Show that integral equals expression

A question is this type if and only if it asks to prove or show that a definite integral equals a specific exact value or expression, requiring full working with integration by parts.

12 questions · Standard +0.2

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CAIE P3 2007 November Q3
4 marks Standard +0.3
3 Use integration by parts to show that $$\int _ { 2 } ^ { 4 } \ln x \mathrm {~d} x = 6 \ln 2 - 2$$
OCR MEI C3 2006 June Q2
6 marks Standard +0.3
2 Show that \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } x \sin 2 x \mathrm {~d} x = \frac { 3 \sqrt { 3 } - \pi } { 24 }\).
OCR MEI C3 Q4
5 marks Standard +0.3
4 Show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos \frac { 1 } { 2 } x \mathrm {~d} x = \frac { \sqrt { 2 } } { 2 } \pi + 2 \sqrt { 2 } - 4\).
[0pt] [5]
OCR MEI C3 Q2
6 marks Standard +0.3
2 Show that \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } x \sin 2 x \mathrm {~d} x = \frac { 3 \sqrt { 3 } \pi } { 24 }\).
OCR C4 Q2
5 marks Moderate -0.3
2. Show that $$\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x = 2 \ln 2 - \frac { 3 } { 4 }$$
OCR MEI C3 2012 January Q3
5 marks Standard +0.3
3 Show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos \frac { 1 } { 2 } x \mathrm {~d} x = \frac { \sqrt { 2 } } { 2 } \pi + 2 \sqrt { 2 } - 4\).
OCR C4 2011 January Q7
7 marks Standard +0.8
7 Show that \(\int _ { 0 } ^ { \pi } \left( x ^ { 2 } + 5 x + 7 \right) \sin x \mathrm {~d} x = \pi ^ { 2 } + 5 \pi + 10\).
OCR H240/01 2019 June Q11
10 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{05bec6d6-b526-4b6f-86f3-39aa38cbf5c6-7_540_734_260_667} The diagram shows part of the curve \(y = \ln ( x - 4 )\).
  1. Use integration by parts to show that \(\int \ln ( x - 4 ) \mathrm { d } x = ( x - 4 ) \ln | x - 4 | - x + c\).
  2. State the equation of the vertical asymptote to the curve \(y = \ln ( x - 4 )\).
  3. Find the total area enclosed by the curve \(y = \ln ( x - 4 )\), the \(x\)-axis and the lines \(x = 4.5\) and \(x = 7\). Give your answer in the form \(a \ln 3 + b \ln 2 + c\) where \(a , b\) and \(c\) are constants to be found.
Edexcel Paper 1 2022 June Q12
5 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
Show that $$\int _ { 1 } ^ { \mathrm { e } ^ { 2 } } x ^ { 3 } \ln x \mathrm {~d} x = a \mathrm { e } ^ { 8 } + b$$ where \(a\) and \(b\) are rational constants to be found.
Edexcel C4 Q1
6 marks Moderate -0.3
  1. Use integration by parts to show that
$$\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x = 2 \ln 2 - \frac { 3 } { 4 }$$
AQA Paper 1 2023 June Q8
6 marks Moderate -0.3
8 Show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } ( x \sin 4 x ) \mathrm { d } x = - \frac { \pi } { 8 }$$
\includegraphics[max width=\textwidth, alt={}]{6a03a035-ff32-4734-864b-a076aa9cbec0-09_2491_1716_219_153}
AQA Paper 3 2021 June Q8
6 marks Standard +0.3
8 Given that $$\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } x \cos x d x = a \pi + b$$ find the exact value of \(a\) and the exact value of \(b\). Fully justify your answer.
[0pt] [6 marks]