Question

The value of ${\int }_0^{\pi }\left[\cos x\right]dx$, where $[\cdot ]$is the greatest integer function, is

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

Answer 1

Answer: (D)

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

$I={\int }_0^{\pi }\left[cosx\right]dx={\int }_0^{\frac{\pi }{2}}\left[cosx\right]dx+{\int }_{\frac{\pi }{2}}^{\pi }\left[cosx\right]dx$
For $0<x<\frac{\pi }{2}\Rightarrow 0<cosx<1\Rightarrow \left[cosx\right]=0$
And for $\frac{\pi }{2}<x<\pi \Rightarrow -1<cosx<0\Rightarrow \left[cosx\right]=-1$
$I={\int }_0^{\frac{\pi }{2}}0dx+{\int }_{\frac{\pi }{2}}^{\pi }(-1)dx={\int }_{\frac{\pi }{2}}^{\pi }(-1)dx={-\left[x\right]}_{\frac{\pi }{2}}^{\pi }=-(\pi -\frac{\pi }{2})=-\frac{\pi }{2}$