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 $u = x - 2$ , find $\displaystyle\int x(x - 2)^5\,dx$ .
Level 4 · 4 marksTry it
2
Using the substitution $u = x - 1$ , evaluate $\displaystyle\int_1^2 x\sqrt{x - 1}\,dx$ .
Level 4 · 4 marksTry it
3
Using the substitution $u = e^x + 1$ , find $\displaystyle\int \frac{e^x}{e^x + 1}\,dx$ .
Level 4 · 3 marksTry it
4
Using the substitution $u = \ln x$ , find $\displaystyle\int \frac{\ln x}{x}\,dx$ .
Level 4 · 3 marksTry it
5
Using the substitution $u = x^{2} + 2$ , find $\displaystyle\int x^{3}\sqrt{x^{2} + 2}\,dx$ . (6 marks)
Level 4 · 6 marksTry it

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