Q. $\int\limits^{\pi}_{{0}}[cot\,x]dx, [•]$ denotes the greatest integer function, is equal to

AIEEEAIEEE 2009Integrals Report Error

A

$\frac{\pi}{2}$

B

$1$

C

$-1$

D

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

Solution:

Let $I=\int\limits^{\pi}_{{0}}[cot\,x]dx \,...(1)$
$=\int\limits^{\pi}_{{0}}[cot(\pi-x)]dx\int\limits^{\pi}_{{0}}--cot\,x]dx \,...(1)$
Adding (1) and (2)
$2I=\int\limits^{\pi}_{{0}}[cot\,x]dx+\int\limits^{\pi}_{{0}}[cot\,x]dx=\int\limits^{\pi}_{{0}}(-1)dx\,\,\,$$\left[\because\left[x\right]+\left[-x\right]=-1 \,if\,x\notin Z=0\,if\,x\in Z\right]$
$=\left[-x\right]^{\pi}_{0}=-\pi$
$\therefore I=-\frac{\pi}{2}$

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