Question

\[\int\frac{1}{\sqrt{1 - x^2} \left( \sin^{- 1} x \right)^2} dx\]

Answer 1

 

\[\int\frac{dx}{\sqrt{1 - x^2} \left( \sin^{- 1} x \right)^2}\]

 

\[Let, \sin^{- 1} x = t\]

\[ \Rightarrow \frac{1}{\sqrt{1 - x^2}} = \frac{dt}{dx}\]

\[ \Rightarrow \frac{1}{\sqrt{1 - x^2}} dx = dt\]

\[Now, \int\frac{dx}{\sqrt{1 - x^2} \left( \sin^{- 1} x \right)^2}\]

\[ = \int\frac{dt}{t^2}\]

\[ = \int t^{- 2} dt\]

\[ = \frac{t^{- 2 + 1}}{- 2 + 1} + C\]

\[ = \frac{- 1}{t} + C\]

\[ = - \frac{1}{\sin^{- 1} x} + C\]