CAIE A-Level Mathematics 3.5.4 An integrand of the form x

Practise recognising f'(x)/f(x) structures in trig and rational integrands, then integrating to logarithmic exact values.

Exam points

recognise f'(x)/f(x) patterns before integrating to a logarithmic expression
use exact trigonometric limits to simplify definite integral answers

[Maximum number: 4]

Let $\mathrm{f}(x)=\frac{\cos x}{1+\sin x}$ .

Find $\int_{\frac{1}{6} \pi}^{\frac{1}{2} \pi} \mathrm{f}(x) \mathrm{d} x$ . Give your answer in a simplified exact form.

State integral of the form $a \ln (1+\sin x)$

*M1

State correct integral $\ln (1+\sin x)$

A1

Use limits correctly

DM1

Obtain answer $\ln \frac{4}{3}$

A1

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3.5.4—An integrand of the form x question 1