Question

If [ ] denotes the greatest integer function then the integral ${\int }_0^{\pi }\left[cosx\right]dx$ is equal to :

• A. $\frac{\pi }{2}$
• B. $0$
• C. $-1$
• D. $-\frac{\pi }{2}$

Answer 1

The correct option is D $-\frac{\pi }{2}$

We know that for $\cos x\in [0,\frac{\pi }{2}]\Rightarrow 1\ge \cos x\ge 0$ and $\cos x\in [\frac{\pi }{2},\pi ]\Rightarrow 0\ge \cos x\ge -1$

${\int }_0^{\pi }[\cos x]dx={\int }_0^{\frac{\pi }{2}}0dx+{\int }_{\frac{\pi }{2}}^{\pi }-1dx$

$=-\frac{\pi }{2}$