The value of $\underset{\pi }{\overset{2\pi }{\int }}\left[2\sin x\right]dx$ where [.] represents the greatest integer function is 

$$\begin{gathered} \mathrm{I}={\int }_{\mathrm{\pi }}^{2\mathrm{\pi }} \left[2 \sin \mathrm{x}\right] \mathrm{dx} \\ \mathrm{For} \mathrm{x}\in \left[\mathrm{\pi }, \frac{7\mathrm{\pi }}{6}\right] \mathrm{and} \left[\frac{11\mathrm{\pi }}{6}, 2\mathrm{\pi }\right] \\ \left[2 \sin \mathrm{x}\right]=-1 \\ \mathrm{For} \mathrm{x}\in \left(\frac{7\mathrm{\pi }}{6}, \frac{11\mathrm{\pi }}{6}\right) \\ \Rightarrow \left[2 \sin \mathrm{x}\right]=-2 \end{gathered}$$

$$\begin{gathered} \therefore \mathrm{I}={\int }_{\mathrm{\pi }}^{7\mathrm{\pi }/6} (-1) \mathrm{dx}+{\int }_{7\mathrm{\pi }/6}^{11\mathrm{\pi }/6} (-2) \mathrm{dx}+{\int }_{11\mathrm{\pi }/6}^{2\mathrm{\pi }} (-1) \mathrm{dx} \\ ={(-1)}_{\mathrm{\pi }}^{\raisebox{1ex}{$7\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}+{(-2)}_{7\mathrm{\pi }/6}^{\raisebox{1ex}{$11\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}+{(-1)}_{\mathrm{\pi \pi }/6}^{2\mathrm{\pi }} \\ =\raisebox{1ex}{$-5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right. \end{gathered}$$