The value of ${\int }_0^{\pi /3}\left[\sqrt{3}\tan x\right]dx$ (where$\left[.\right]$ denotes the greatest integer function )

$\begin{array}{rcl}\mathrm{I}& =& {\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} \left[\sqrt{3} \tan \mathrm{x}\right]\\ 0& <& \mathrm{x}<\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\\ 0& <& \tan \mathrm{x}<\sqrt{3}\\ 0& <& \sqrt{3} \tan \mathrm{x}<3\\ \sqrt{3} \tan \mathrm{x}& =& 0, 1, 2, 3, 4, ...\\ \tan \mathrm{x}& =& 0, \frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}, \sqrt{3}, ...\\ \mathrm{x}& =& 0, {\tan }^{-1}\frac{1}{\sqrt{3}}, {\tan }^{-1}\frac{2}{\sqrt{3}}, {\tan }^{-1}\sqrt{3}\\ & =& 0, \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right., {\tan }^{-1}\frac{2}{\sqrt{3}}, \frac{\mathrm{\pi }}{3}\end{array}$

$\begin{array}{rcl}\mathrm{I}& =& {\int }_0^{\frac{\pi }{6}} 0 \mathrm{dx}+{\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}^{{\tan }^{-1} \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.} 1 \mathrm{dx}+{\int }_{{\tan }^{-2} \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3 $}\right.}2 \mathrm{dx}\\ & =& 0+\left({\tan }^{-1} \frac{2}{\sqrt{3}}-\frac{\mathrm{\pi }}{6}\right)+2\left(\frac{\mathrm{\pi }}{3}-{\tan }^{-1}\frac{2}{\sqrt{3}}\right)\\ & =& \frac{\mathrm{\pi }}{2}-{\tan }^{-1}\frac{2}{\sqrt{3}}\end{array}$