Basic integration by parts

A question is this type if and only if it asks to find or evaluate a single definite or indefinite integral using integration by parts once, with standard functions like x·e^(ax), x·sin(ax), x·cos(ax), or x·ln(x).

27 questions · Moderate -0.2

1.08i Integration by parts
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CAIE P3 2003 June Q2
4 marks Moderate -0.5
2 Find the exact value of \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x\).
CAIE P3 2011 June Q3
5 marks Moderate -0.3
3 Show that \(\int _ { 0 } ^ { 1 } ( 1 - x ) \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x = 4 \mathrm { e } ^ { - \frac { 1 } { 2 } } - 2\).
CAIE P3 2016 June Q2
5 marks Moderate -0.8
2 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
CAIE P3 2017 June Q4
4 marks Moderate -0.3
4 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \theta \sin \frac { 1 } { 2 } \theta \mathrm {~d} \theta\).
CAIE P3 2020 June Q2
5 marks Moderate -0.3
2 Find the exact value of \(\int _ { 0 } ^ { 1 } ( 2 - x ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
CAIE P3 2020 March Q4
7 marks Moderate -0.3
4 Find \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } x \sec ^ { 2 } x \mathrm {~d} x\). Give your answer in a simplified exact form.
CAIE P3 2021 November Q4
5 marks Moderate -0.3
4 Find the exact value of \(\int _ { \frac { 1 } { 3 } \pi } ^ { \pi } x \sin \frac { 1 } { 2 } x \mathrm {~d} x\).
CAIE P3 2022 November Q3
5 marks Standard +0.3
3 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \sec ^ { 2 } x \mathrm {~d} x\).
Edexcel C4 2011 January Q1
6 marks Moderate -0.3
  1. Use integration to find the exact value of
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \sin 2 x \mathrm {~d} x$$
OCR MEI C3 2008 June Q2
4 marks Moderate -0.5
2 Find \(\int x \mathrm { e } ^ { 3 x } \mathrm {~d} x\).
OCR MEI C3 Q5
5 marks Moderate -0.3
5 Find \(\int _ { 2 } ^ { 3 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x\), giving your answer to 1 decimal place.
OCR C4 Specimen Q3
5 marks Moderate -0.3
3 Find \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), giving your answer in terms of e.
Edexcel C34 2016 June Q5
6 marks Standard +0.3
5.Use integration by parts to find the exact value of $$\int _ { 0 } ^ { 2 } x 2 ^ { x } \mathrm {~d} x$$ Write your answer as a single simplified fraction.
OCR C4 2009 January Q2
4 marks Standard +0.3
2 Find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).
OCR C4 2013 January Q1
4 marks Moderate -0.3
1 Find \(\int x \cos 3 x \mathrm {~d} x\).
OCR MEI Paper 3 2021 November Q7
3 marks Moderate -0.3
7 Determine \(\int x \cos 2 x \mathrm {~d} x\).
Edexcel C4 Q2
7 marks Moderate -0.3
2. (a) Use integration by parts to find $$\int x \cos 2 x d x$$ (b) Prove that the answer to part (a) may be expressed as $$\frac { 1 } { 2 } \sin x ( 2 x \cos x - \sin x ) + C ,$$ where \(C\) is an arbitrary constant.
Pre-U Pre-U 9794/1 2013 November Q9
Moderate -0.5
9 Find \(\int x \sin 2 x \mathrm {~d} x\).
Edexcel P4 2024 June Q1
5 marks Standard +0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Find $$\int_0^{\pi/6} x \cos 3x \, dx$$ giving your answer in simplest form. [5]
Edexcel C4 Q18
7 marks Moderate -0.3
  1. Use integration by parts to find $$\int x \cos 2x \, dx.$$ [4]
  2. Prove that the answer to part \((a)\) may be expressed as $$\frac{1}{2} \sin x (2x \cos x - \sin x) + C,$$ where \(C\) is an arbitrary constant. [3]
OCR MEI C3 Q4
4 marks Standard +0.3
Find \(\int x \sin 3x dx\). [4]
OCR MEI C3 Q4
4 marks Moderate -0.3
Find \(\int xe^{3x} \, dx\). [4]
OCR C4 2007 January Q2
5 marks Standard +0.3
Find the exact value of \(\int_1^2 x \ln x \, dx\). [5]
OCR C4 2005 June Q2
5 marks Moderate -0.3
Evaluate \(\int_0^{\frac{\pi}{2}} x \cos x dx\), giving your answer in an exact form. [5]
OCR C4 Q1
4 marks Moderate -0.5
Find \(\int xe^{3x} dx\). [4]