Question

Q Integrate the following
$\int {\sin }^{-1}\left(\cos x\right)$

Answer 1

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