Question

$\int {\sin }^{-1}\left(\cos x\right)dx$

Answer 1

$\begin{array}{c}\int {\sin }^{-1}\left(\cos x\right)dx=\int {\sin }^{-1}\left\{\sin \left(\frac{\pi }{2}-x\right)\right\}dx\\ =\int \left(\frac{\pi }{2}-x\right)dx\\ =\frac{\pi }{2}.\int dx-\int xdx\\ =\frac{\pi x}{2}-\frac{x^2}{2}+C.\end{array}$

Hence, this is the answer.