2022TrialShort answer

Q12 (d) (ii)

2 marks

Question

Find $\int\sqrt{\frac{x-1}{x+1}}\,dx.$

Try the question before revealing the answer.

Solution

$\int\sqrt{\frac{x-1}{x+1}}\,dx$

$=\int\frac{x-1}{\sqrt{x^2-1}}\,dx$

$=\int\frac{x}{\sqrt{x^2-1}}\,dx-\int\frac{1}{\sqrt{x^2-1}}\,dx$

$=\frac12\int\frac{2x}{\sqrt{x^2-1}}\,dx-\int\frac{1}{\sqrt{x^2-1}}\,dx$

$=\sqrt{x^2-1}-\ln\left(x+\sqrt{x^2-1}\right)+C.$

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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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