2020HSCShort answer

Q13 (d) (iii)

2 marks

Question

Hence, or otherwise, find
$\int_0^{\pi/2}\cos^4\theta\,d\theta.$

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
Uses part (a) and attempts to integrate, or equivalent merit	1

Sample answer:

\int_0^{\pi/2}\cos^4\theta\,d\theta

=\int_0^{\pi/2}\frac18(\cos4\theta+4\cos2\theta+3)\,d\theta

=\frac18\left(\frac{\sin4\theta}{4}+2\sin2\theta+3\theta\right)_0^{\pi/2}

=\frac18\left(\frac{3\pi}{2}\right)=\frac{3\pi}{16}

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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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