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.

Start practising Read the lesson

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
(a) Find

$\int x\sin 3x\,dx$

(4 marks)
(b) Hence find the particular solution of the differential equation

$\dfrac{dy}{dx} = \dfrac{9x\sin 3x}{y^2}$

for which $y = 2$ when $x = 0$ . Give your answer in the form $y^n = \text{f}(x)$ , stating the value of the integer $n$ .

(5 marks)
Level 4 · 9 marksTry it
2
The region $R$ is bounded by the curve $y = \dfrac{\ln x}{x^2}$ , the $x$ -axis and the line $x = \mathrm{e}$ .

Find the exact area of $R$ . (5 marks)
Level 4 · 5 marksTry it
3
(a) Find $\displaystyle\int x\sin 3x\;\mathrm{d}x$ . (4 marks)
(b) The region $R$ is bounded by the curve $y = x\sin 3x$ , the $x$ -axis and the line $x = \dfrac{\pi}{3}$ .

Find the exact area of $R$ . (4 marks)
Level 4 · 8 marksTry it
4
(a) Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(\sec x + \tan x) = \sec x$ .

(2 marks)
(b) Hence show that $\displaystyle\int_{0}^{\frac{1}{3}\pi} x\sec x\tan x\,\mathrm{d}x = \tfrac{2}{3}\pi - \ln\left(2 + \sqrt{3}\right)$ .

(5 marks)
Level 4 · 7 marksTry it
5
(a) Use integration by parts to find

$\displaystyle\int x^3 \ln x\; dx$ . (5 marks)
(b) Hence show that

$\displaystyle\int_1^e x^3 \ln x\; dx = \frac{3e^4 + 1}{16}$ . (3 marks)
Level 4 · 8 marksTry it

4 more questions on integration by parts

Free with an account — just an email, no card. The whole bank, every mark scheme and worked solution.

Start free