2022HSCShort answer

Q11 (f)

3 marks

Question

Using the substitution $t=\tan\frac{x}{2}$ , find
$\int\frac{dx}{1+\cos x-\sin x}.$

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	3
Obtains integrand in terms of $t$ in simplest form, or equivalent merit	2
Correctly replaces $dx$ in terms of $t$ and $dt$ , or equivalent merit	1

Sample answer:

t=\tan\frac{x}{2}\quad\Rightarrow\quad dx=\frac{2\,dt}{1+t^2}

\int\frac{dx}{1+\cos x-\sin x} =\int\frac{\frac{2\,dt}{1+t^2}}{1+\frac{1-t^2}{1+t^2}-\frac{2t}{1+t^2}} =\int\frac{2\,dt}{2-2t} =\int\frac{dt}{1-t} =-\ln|1-t|+k =-\ln\left|1-\tan\frac{x}{2}\right|+k

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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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