Question

The value of $\frac{6}{\pi }\stackrel{\pi /3}{\underset{\pi /6}{\int }}\left[2\sin x\right]dx$, where [.] represents greatest integer function is

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is A $1$

Let $I=\frac{6}{\pi }{\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\left[2\sin x\right]dx$

Now, $\sin xϵ(\frac{1}{2},\frac{\sqrt{3}}{2}),xϵ(\frac{\pi }{6},\frac{\pi }{3})$

$\therefore [2\sin x{]}_{\frac{\pi }{6}}^{\frac{\pi }{3}}=1$

$\therefore I=\frac{6}{\pi }{\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}1.dx=\frac{6}{\pi }=\frac{\pi }{6}$

$I=1$