2022TrialShort answer

Q14 (d) (ii)

3 marks

Question

Hence, or otherwise, evaluate
$\int_0^{\pi/4}\frac{\sin x}{\cos x+\sin x}\,dx.$

Try the question before revealing the answer.

Solution

$\int_0^{\pi/4}\frac{\sin x}{\cos x+\sin x}\,dx=\int_0^{\pi/4}\frac{\sin(\frac{\pi}{4}-x)}{\cos(\frac{\pi}{4}-x)+\sin(\frac{\pi}{4}-x)}\,dx$

$=\int_0^{\pi/4}\frac{\cos x-\sin x}{\cos x+\sin x+\cos x-\sin x}\,dx$
$=\int_0^{\pi/4}\frac{\cos x-\sin x}{2\cos x}\,dx$
$=\frac12\int_0^{\pi/4}(1-\tan x)\,dx$
$=\frac12\left[x+\ln(\cos x)\right]_0^{\pi/4}$
$=\frac12\left[\frac\pi4+\ln\left(\frac{1}{\sqrt2}\right)-(0+\ln1)\right]$
$=\frac12\left(\frac\pi4+\ln\left(\frac{1}{\sqrt2}\right)\right)$

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Syllabus alignment

Calculus

Further IntegrationApply substitution, partial fractions, integration by parts and recurrence relationships to theoretical and practical problems.

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