MCQ
If $[\,\,]$ denotes the greatest integer function, then the integral $\int\limits_0^\pi {[\cos \,\,x\,\,dx]} $ is equal
- A$\frac {\pi }{2}$
- B$0$
- C$-1$
- ✓$-\frac {\pi }{2}$
$I = \int\limits_0^\pi {\left[ {\cos \left( {\pi - x} \right)} \right]dx} $
$ = \int\limits_0^\pi {\left[ { - \cos x} \right]dx} \,\,\,\,\,......\left( 2 \right)$
On adding $(1)$ and $(2)$, we get
$2I = \int\limits_0^\pi {\left[ {\cos \left( {\pi - x} \right)} \right]dx + } \int\limits_0^\pi {\left[ { - \cos x} \right]dx} $
$2I = \int\limits_0^\pi {\left[ {\cos x} \right] + } \left[ { - \cos x} \right]dx$
$2I = \int\limits_0^\pi { - 1dx} $ ($\because $ $\left[ x \right] + \left[ { - x} \right] = - 1$ if $x \notin {\rm Z}$)
$2I = - \left. x \right|_0^\pi \,\, = - \pi $
$ \Rightarrow I = \frac{{ - \pi }}{2}$
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where $\{x\}$ and $[x]$ denotes the fractional part $\&$ integral part functions.