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

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Solution

$\int \frac{{sin}^{- 1} x}{(1 - x^{2})^{\frac{3}{2}}} d x$
Let $u = {sin}^{- 1} x$
$d u = \frac{1}{(- 1 x^{2})} d x$
$= \frac{{sin}^{1} x}{\sqrt{1 - x^{2}}} - \int \frac{x}{1 - x^{2}} d x + C$ ..... $(1)$
$= \int \frac{x}{1 - x^{2}} d x = \frac{- 1}{2} log | 1 - x^{2} |$
$= \int \frac{x}{1 - x^{2}} d x$
Let $w = 1 - x^{2}$
$d w = - 2 x d x$
$= \frac{1}{2} \int \frac{- 2 x}{1 - x^{2}} d x = \frac{- 1}{2} \int \frac{d w}{w} = \frac{- 1}{2} log | w |$
$= \frac{- 1}{2} log | 1 - x^{2} |$
Now putting this value in eq $(1)$ we get
$= \frac{{sin}^{- 1} x}{\sqrt{1 - x^{2}}} - (\frac{- 1}{2} log | 1 - x^{2} |)$
$= \frac{{sin}^{- 1} x}{\sqrt{1 - x^{2}}} + \frac{1}{2} log | 1 - x^{2} | + C$

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