A-level · Integration

Integration by substitutionExam-style questions

Integrating by changing the variable — for integrals recognition cannot do. Choosing u to rename the awkward part; replacing dx with du; making every x disappear, including the strays; integrating in u and converting back; changing the limits of a definite integral so nothing converts back; and choosing a substitution, including a root or a trig function.

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32 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
Using the substitution $u = 2x + 1$ , find $\displaystyle\int x\sqrt{2x + 1}\,dx$ .
Level 3 · 4 marksTry it
2
Using the substitution $u = x^2 + 1$ , evaluate $\displaystyle\int_0^1 \frac{x}{(x^2 + 1)^2}\,dx$ .
Level 3 · 4 marksTry it
3
Using a suitable substitution, find $\displaystyle\int x^2(x^3 + 1)^5\,dx$ .
Level 3 · 3 marksTry it
4
Using the substitution $u = x + 3$ , find $\displaystyle\int x\sqrt{x + 3}\,dx$ . (5 marks)
Level 3 · 5 marksTry it
5
Using the substitution $u = \sin x$ , evaluate $\displaystyle\int_{0}^{\pi/2} \cos x\sin^{4} x\,dx$ . (4 marks)
Level 3 · 4 marksTry it

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