2022TrialShort answer

Q12 (e)

3 marks

Question

Use the substitution $x=2\sin\theta$ to find
$\int\frac{dx}{(4-x^2)^{3/2}}.$

Try the question before revealing the answer.

Solution

\int\frac{dx}{(4-x^2)^{3/2}} =\int\frac{2\cos\theta}{(4-4\sin^2\theta)^{3/2}}\,d\theta

=\int\frac{2\cos\theta}{8\cos^3\theta}\,d\theta

=\frac14\int\frac{1}{\cos^2\theta}\,d\theta

=\frac14\int\sec^2\theta\,d\theta

=\frac14\tan\theta+C

Using $x=2\sin\theta$ ,

\tan\theta=\frac{x}{\sqrt{4-x^2}}

Therefore,

\frac{x}{4\sqrt{4-x^2}}+C

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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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