Double integration by parts

A question is this type if and only if it requires applying integration by parts twice to evaluate integrals like x²·sin(x), x²·cos(x), or x²·e^(ax).

10 questions · Standard +0.3

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CAIE P3 2010 June Q2
5 marks Standard +0.3
2 Show that \(\int _ { 0 } ^ { \pi } x ^ { 2 } \sin x \mathrm {~d} x = \pi ^ { 2 } - 4\).
CAIE P3 2016 June Q3
5 marks Standard +0.3
3 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { 2 } \sin 2 x \mathrm {~d} x\).
CAIE P3 2019 June Q2
5 marks Standard +0.3
2 Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x ^ { 2 } \cos 2 x \mathrm {~d} x = \frac { 1 } { 32 } \left( \pi ^ { 2 } - 8 \right)\).
Edexcel C4 2013 June Q1
7 marks Standard +0.3
  1. Find \(\int x ^ { 2 } e ^ { x } d x\).
  2. Hence find the exact value of \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x\).
OCR C4 2007 June Q2
6 marks Standard +0.3
2 Find the exact value of \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x\).
OCR C4 2012 January Q9
7 marks Standard +0.3
9 Find the exact value of \(\int _ { 0 } ^ { 1 } \left( x ^ { 2 } + 1 \right) \mathrm { e } ^ { 2 x } \mathrm {~d} x\).
OCR MEI Paper 1 Specimen Q8
7 marks Standard +0.3
8 Find \(\int x ^ { 2 } \mathrm { e } ^ { 2 x } \mathrm {~d} x\).
OCR MEI Paper 2 2023 June Q10
5 marks Standard +0.3
10 Determine the exact value of \(\int _ { 0 } ^ { \frac { \pi } { 4 } } 4 x \cos 2 x d x\).
Edexcel C4 Q1
6 marks Standard +0.8
  1. Use integration by parts to find
$$\int x ^ { 2 } \sin x d x$$
Edexcel C4 Q2
7 marks Standard +0.3
2. Use integration by parts to find $$\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$