2022TrialShort answer

Q16 (a) (iii)

2 marks

Question

Hence find
$\int_0^{\pi/2}\cos^7\theta\,d\theta.$

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Solution

(iii)
$\int_0^{\frac{\pi}{2}}\cos^7\theta\,d\theta=\frac{1}{64}\int_0^{\frac{\pi}{2}}\left(\cos7\theta+7\cos5\theta+21\cos3\theta+35\cos\theta\right)d\theta$
$=\frac{1}{64}\left[\frac{1}{7}\sin7\theta+\frac{7}{5}\sin5\theta+7\sin3\theta+35\sin\theta\right]_0^{\frac{\pi}{2}}$
$=\frac{1}{64}\left(-\frac{1}{7}+\frac{7}{5}-7+35\right)$
$=\frac{1}{64}\times\frac{1024}{35}=\frac{16}{35}$

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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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