First slide

Methods of integration

Question

$\int \frac{\sin^{- 1} x}{{(1 - x^{2})}^{3 / 2}} dx$ is equal to

Easy

A

$(\sin^{- 1} x) \frac{x}{\sqrt{1 - x^{2}}} + \log (\sqrt{1 - x^{2}}) + C$
B

$\frac{\sin^{- 1} x}{\sqrt{1 - x^{2}}} + \log (\sqrt{1 - x^{2}}) + C$
C

$\frac{{xsin}^{- 1} x}{\sqrt{1 - x^{2}}} - \log (\sqrt{1 - x^{2}}) + C$
D

None of the above

Solution

$Let I = \int \frac{\sin^{- 1} x}{(1 - x^{2}) \sqrt{1 - x^{2}}} dx$
$Put \sin^{- 1} x = t \Rightarrow \frac{1}{\sqrt{1 - x^{2}}} = \frac{dt}{dx} \Rightarrow dx = \sqrt{1 - x^{2}} dt$ $∴ I = \int \frac{t}{(1 - \sin^{2} t) \sqrt{1 - x^{2}}} \cdot \sqrt{1 - x^{2}} dt$
$= \int \frac{t}{(1 - \sin^{2} t)} dt = \int_{} {tsec}^{2} dt [∵ 1 - \sin^{2} t = \cos^{2} t]$
$= t \int \sec^{2} tdt - \int [\frac{d}{dt} t \int \sec^{2} tdt] dt$
[integration by parts]
$= ttan t - \int \tan tdt = ttan t + \log \cos t + C$
$= (\sin^{- 1} x) \frac{x}{\sqrt{1 - x^{2}}} + \log (\sqrt{1 - x^{2}}) + C$
$[fromfigure \tan t = \frac{x}{\sqrt{1 - x^{2}}} and \cos t = \sqrt{1 - x^{2}}]$

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