A-level · Integration

Integration by partsExam-style questions

Integrating a product of two unrelated factors by reversing the product rule. Where the formula comes from; choosing which factor to differentiate, so the integral left over is easier; integrating ln x by writing it as ln x × 1; applying parts twice when a power of x needs two passes; and definite integrals by parts, with exact answers.

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27 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
(a) Using integration by parts, find the exact value of

$\int_0^{\frac{\pi}{4}} x^2\cos 2x\,dx$

writing your answer in its simplest form.

(5 marks)
(b) Using a suitable substitution, show that

$\int_1^5 \dfrac{x}{1 + \sqrt{2x - 1}}\,dx = a + b\ln 2$

where $a$ and $b$ are rational constants to be found.

(7 marks)
Level 5 · 12 marksTry it
2
(a) Use integration by parts to show that $\displaystyle\int \ln x\,\mathrm{d}x = x\ln x - x + c$ . (3 marks)
(b) Hence find the exact area of the region bounded by $y = \ln x$ , the $x$ -axis and the line $x = e$ . (4 marks)
Level 5 · 7 marksTry it
3
(a) Find $\displaystyle\int x e^{x}\,\mathrm{d}x$ . (3 marks)
(b) Hence find the exact area of the region bounded by $y = x e^{x}$ , the $x$ -axis and the lines $x = 0$ and $x = 1$ . (4 marks)
Level 5 · 7 marksTry it
4
In this question you must show detailed reasoning.

The curve $C$ has equation $y = x^2\ln x$ , for $x > 0$ .

The region $R$ is bounded by $C$ , the $x$ -axis and the line $x = \mathrm{e}$ .

Show that the area of $R$ is $\dfrac{2\mathrm{e}^3 + 1}{9}$ . (8 marks)
Level 5 · 8 marksTry it
5
(a) Find $\displaystyle\int x\cos 3x\,\mathrm{d}x$ . (3 marks)

The curve $C$ has equation $y = x\cos 3x$ , $\ 0 \le x \le \dfrac{\pi}{2}$ .
(b) Find the exact value of $\displaystyle\int_0^{\frac{\pi}{2}} x\cos 3x\,\mathrm{d}x$ . (2 marks)
(c) Explain why your answer to part (b) is not the total area of the regions enclosed by $C$ and the $x$ -axis for $0 \le x \le \dfrac{\pi}{2}$ , and find this total area, giving your answer in exact form. (4 marks)
Level 5 · 9 marksTry it

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