Question

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

• A. $\frac{-5\pi }{3}$
• B. $-\pi$
• C. $\frac{5\pi }{3}$
• D. $-2\pi$

Answer 1

The correct option is A $\frac{-5\pi }{3}$

${\int }_{\pi }^{2\pi }\left[2\sin x\right]dx={\int }_{\pi }^{7\pi /6}\left[2\sin x\right]dx+{\int }_{7\pi /6}^{11\pi /6}\left[2\sin x\right]dx+{\int }_{11\pi /6}^{2\pi }\left[2\sin x\right]dx$
$={\int }_{\pi }^{7\pi /6}(-1)dx+{\int }_{7\pi /6}^{11\pi /6}(-2)dx+{\int }_{11\pi /6}^{2\pi }(-1)dx$
$=-\frac{\pi }{6}-2\left(\frac{4\pi }{6}\right)-\frac{\pi }{6}=-\frac{10\pi }{6}=-\frac{5\pi }{3}$