A-level · Integration

Integration by reversing the chain ruleExam-style questions

Integrating by recognition — spotting the chain rule's fingerprint, the derivative of an inside function sitting as a factor. A power of a function; adjusting the answer when the factor is a number out; the log case, when the top is the derivative of the bottom; exponential and trig insides; and how to tell when recognition fails and you need to expand or substitute.

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19 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
Find $\displaystyle\int 2x(x^2 + 1)^3\,dx$ .
Level 2 · 2 marksTry it
2
Find $\displaystyle\int \frac{6x^2}{x^3 + 4}\,dx$ .
Level 2 · 2 marksTry it
3
Find $\displaystyle\int 3x^{2}(x^{3} - 2)^{5}\,dx$ . (3 marks)
Level 2 · 3 marksTry it
4
(a) Find $\dfrac{d}{dx}\left(5 - 2x^2\right)^4$

(2 marks)
(b) Hence show that

$\int_0^1 x\left(5 - 2x^2\right)^3 dx = R$

where $R$ is an integer to be found.

(Solutions relying on calculator technology are not acceptable.)

(3 marks)
Level 3 · 5 marksTry it
5
Find $\displaystyle\int x\sqrt{x^2 + 5}\,dx$ .
Level 3 · 2 marksTry it

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