Question

$\int \frac{1}{\sqrt{\left(1-x^2\right)\left\{9+{\left({\sin }^{-1}x\right)}^2\right\}}}dx$

Answer 1

$$\begin{gathered} \int \frac{dx}{\sqrt{\left(1-x^2\right)\left(9+{\left({\sin }^{-1}x\right)}^2\right)}} \\ \mathrm{let}{\sin }^{-1}x=t \\ \Rightarrow \frac{1}{\sqrt{1-x^2}}dx=dt \\ \mathrm{Now},\int \frac{dx}{\sqrt{\left(1-x^2\right)\left(9+{\left({\sin }^{-1}x\right)}^2\right)}} \\ =\int \frac{dt}{\sqrt{9+t^2}} \\ =\int \frac{dt}{\sqrt{3^2+t^2}} \\ =\log \left|t+\sqrt{3^2+t^2}\right|+C \\ =\log \left|{\sin }^{-1}x+\sqrt{9+{\left({\sin }^{-1}x\right)}^2}\right|+C \end{gathered}$$