Question

Evaluate: ${\int }_0^{\pi /3}\left[\sqrt{3}\tan x\right]dx$

where [.] is Greatest integer function

• A. $\frac{5\pi }{6}$
• B. $\frac{5\pi }{6}-{\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$
• C. $\frac{\pi }{2}-{\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (C)

$\frac{\pi }{2}-{\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

${\int }_0^{\frac{\pi }{3}}\left[\sqrt{3}\tan x\right]dx={\int }_0^{\frac{\pi }{6}}1dx+{\int }_{\frac{\pi }{6}}^{\frac{\pi }{4}}0dx+{\int }_{\frac{\pi }{4}}^{\frac{\pi }{3}}1-2dx=\frac{\pi }{6}-[\frac{\pi }{3}-\frac{\pi }{4}]=\frac{\pi }{6}-\frac{\pi }{12}=\frac{\pi }{12}={\cot }^{-1}\left(\frac{2}{\sqrt{3}}\right)=\frac{\pi }{2}-{\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$