Question

If the value of the definite integral $A={\int }_0^{10\pi }\left[sinx\right]dx$ is equal to $k\pi ,$ then the absolute value of $k$ is equal to (where, $\left[\cdot \right]$ is the greatest integer function)

Answer 1

Answer: 5

As period of $sinx$ is $2\pi ,$
$I=\left(5\right){\int }_0^{2\pi }\left[sinx\right]dx$
$I=\left(5\right)\left({\int }_0^{\pi }0dx+{\int }_{\pi }^{2\pi }\left(-1\right)dx\right)$
$=5{\left[-x\right]}_{\pi }^{2\pi }$
$=-5\pi$
$\therefore k=-5\Rightarrow \left|k\right|=5$