Question

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

Answer 1

$$\begin{gathered} \int \frac{dx}{\sqrt{1-x^2}{\left({\sin }^{-1}x\right)}^2} \\ \mathrm{Let},{\sin }^{-1}x=t \\ \Rightarrow \frac{1}{\sqrt{1-x^2}}=\frac{dt}{dx} \\ \Rightarrow \frac{1}{\sqrt{1-x^2}}dx=dt \\ \mathrm{Now},\int \frac{dx}{\sqrt{1-x^2}{\left({\sin }^{-1}x\right)}^2} \\ =\int \frac{dt}{t^2} \\ =\int t^{-2}dt \\ =\frac{t^{-2+1}}{-2+1}+\mathrm{C} \\ =\frac{-1}{t}+\mathrm{C} \\ =-\frac{1}{{\sin }^{-1}x}+\mathrm{C} \end{gathered}$$