$\int \frac{s i n^{- 1} x}{(1 - x^{2})^{3 / 4}} d x$

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Solution

Let $I = \int \frac{s i n^{- 1} x}{(1 - x^{2})^{3 / 4}} d x = \int \frac{s i n^{- 1} x}{(1 - x^{2}) \sqrt{1 - x^{2}}} d x P u t s i n^{- 1} x = t \Rightarrow \frac{1}{\sqrt{1 - x^{2}}} d x = d t a n d x = s i n t \Rightarrow 1 - x^{2} = c o s^{2} t \Rightarrow c o s t = \sqrt{1 - x^{2}} ∴ I = \int \frac{t}{c o s^{2} t} d t$

Using integration by parts,
$I = t \int s e c^{2} t d t - \int (\frac{d}{d t} t . \int s e c^{2} t d t) d t = t . t a n t - \int t a n t d t = t . t a n t - \int 1. t a n t d t = t . t a n t + l o g | c o s t | + C [∵ \int t a n x d x = - l o g | c o s x | + C] = s i n^{- 1} x . \frac{x}{\sqrt{1 - x^{2}}} + l o g | \sqrt{1 - x^{2}} | C$

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