Question

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

Answer 1

$I={\int }_0^x\left[\sin t\right]dt$

$={\int }_0^{2n\pi }\left[\sin t\right]dt+{\int }_{2n\pi }^x\left[\sin t\right]dt$

$=n{\int }_0^{2\pi }\left[\sin t\right]dt+{\int }_{2n\pi }^x\left[\sin t\right]dt$

$=-n\pi +0=-n\pi$ for $x\in (2n\pi ,(2n+1)\pi )$