Question

If the value of the definite integral $A=\displaystyle \int _{0}^{10 \pi } \left[sin x\right] d x$ 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

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