Question

The value of integral $\underset{0}{\overset{2\pi }{\int }}\left[\sin x\right]dx$ is, where $[\cdot ]$ denotes the greatest integer function

• A. $-\pi$
• B. $2\pi$
• C. $\pi$
• D. $4\pi$

Answer 1

The correct option is A $-\pi$

Let $I=\underset{0}{\overset{2\pi }{\int }}\left[\sin x\right]dx$
$\Rightarrow I=\underset{0}{\overset{\pi }{\int }}0dx+$$\underset{\pi }{\overset{2\pi }{\int }}(-1)dx$
$\Rightarrow I=0-1(2\pi -\pi )=-\pi$