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).

22 questions · Moderate -0.3

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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
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \sin 2 x \mathrm {~d} x$$
Edexcel P4 2024 June Q1
5 marks Moderate -0.3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Find $$\int _ { 0 } ^ { \frac { \pi } { 6 } } x \cos 3 x d x$$ giving your answer in simplest form.
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 MEI C3 Q4
4 marks Moderate -0.5
4 Find \(\int x \mathrm { e } ^ { 3 x } \mathrm {~d} x\).
OCR C4 2005 June Q2
5 marks Moderate -0.5
2 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \cos x \mathrm {~d} x\), giving your answer in an exact form.
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.
Edexcel C4 Q5
6 marks Moderate -0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5a77103e-73c1-4f32-af93-f2d5627e2290-02_467_783_317_303}
\end{figure} Figure 1 shows the graph of the curve with equation $$y = x \mathrm { e } ^ { 2 x } , \quad x \geq 0 .$$ The finite region \(R\) bounded by the lines \(x = 1\), the \(x\)-axis and the curve is shown shaded in Figure 1.
  1. Use integration to find the exact value of the area for \(R\).
  2. Complete the table with the values of \(y\) corresponding to \(x = 0.4\) and 0.8 .
    \(x\)00.20.40.60.81
    \(y = x \mathrm { e } ^ { 2 x }\)00.298361.992077.38906
  3. Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures.
Edexcel C4 Q18
7 marks Moderate -0.5
18. (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.
[0pt] [P3 June 2002 Question 2]