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.

Start practising Read the lesson

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
The curve $C$ has equation

$y = \frac{4x}{x^2 + 1} \qquad x \geqslant 0$

The region $R$ is bounded by the curve $C$ , the $x$ -axis and the line with equation $x = k$ , where $k$ is a positive constant.
(a) Show that the area of $R$ is $2\ln\left(k^2 + 1\right)$

(3 marks)

Given that the area of $R$ is $\ln 81$
(b) find the exact value of $k$ .

(2 marks)
Level 4 · 5 marksTry it