Question

${\int }_0^{\infty }\left[{\cos }^{-1}x\right]dx$ (where $[.]$ denotes greatest integer function)

Answer 1

${\int }_0^{\infty }{\cos }^{-1}\left(x\right)dx$

$\int {\cos }^{-1}\left(x\right)dx=x{\cos }^{-1}\left(x\right)-\sqrt{1-x^2}+C$

upon computing boundaries, we get,

${\int }_0^{\infty }{\cos }^{-1}\left(x\right)dx=diverges$