Question
The value of $$\int_0^\pi {\left[ {\cos \,x} \right]dx} ,$$ where $$\left[ . \right]$$ is the greatest integer function, is
A.
$$\frac{\pi }{2}$$
B.
0
C.
$$\pi $$
D.
$$ - \frac{\pi }{2}$$
Answer :
$$ - \frac{\pi }{2}$$
Solution :
$$\eqalign{
& I = \int_0^{\frac{\pi }{2}} {\left[ {\cos \,x} \right]dx} + \int_{\frac{\pi }{2}}^\pi {\left[ {\cos \,x} \right]dx} \cr
& \,\,\,\,\, = \int_0^{\frac{\pi }{2}} {0\,dx} + \int_{\frac{\pi }{2}}^\pi {\left( { - 1} \right)dx} \cr
& \,\,\,\,\, = - \left( {\pi - \frac{\pi }{2}} \right) \cr
& \,\,\,\,\, = - \frac{\pi }{2} \cr} $$