Question

If [.] denotes the greatest integer function then $\underset{\pi }{\overset{2\pi }{\int }}[2\sin x]dx$ is equal to

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

Answer 1

Answer: (C)

$I=\underset{\pi }{\overset{\pi +\pi /6}{\int }}(-1)dx+\underset{\pi +\frac{\pi }{6}}{\overset{\pi +\frac{5\pi }{6}}{\int }}(-2)dx+\underset{\pi +\frac{5\pi }{6}}{\overset{2\pi }{\int }}(-1)dx$
$=-\frac{\pi }{6}-2\left(\frac{4\pi }{6}\right)-\left(\frac{\pi }{6}\right)=-\frac{5\pi }{3}$