Question

Find the integrals of the function :
${\sin }^{-1}\left(\cos x\right)dx$

Answer 1

Consider the given integral.

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

We know that,

$\sin (\frac{\pi }{2}-\theta )=\cos \theta$

${\sin }^{-1}\left(\sin x\right)=x$

Therefore,

$I=\int {\sin }^{-1}\left[\sin (\frac{\pi }{2}-x)\right]dx$

$I=\int (\frac{\pi }{2}-x)dx$

$I=\frac{\pi }{2}\int dx-\int xdx$

$I=\frac{\pi }{2}x-\frac{x^2}{2}+C$

Hence, this is the required value of the integral.