Question

Evaluate ${\int }_0^x\left[\cos t\right]dt$, where $n\in (2n\pi ,(4n+1)\frac{\pi }{2}),n\in N$, and $[.]$ denotes the greatest integer function.

Answer 1

$t\in (2n\pi ,(4n+1)\frac{\pi }{2})$

$\cos t\in (0,1)⟹\left[\cos t\right]=0$

${\int }_0^x\left[\cos t\right]dt=0$