_0^ [ x ]dx = - Vidyadip

MCQ

$\mathop \smallint \limits_0^\pi \left[ {\cot x} \right]dx = $

A
$1$
B
$-1$
✓
$ - \frac{\pi }{2}$
D
$\;\frac{\pi }{2}$

✓

Answer

Correct option: C.

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

c
$I = \int\limits_0^\pi {\left[ {\cot x} \right]} dx$

$I = \int\limits_0^\pi {\left[ {\cot \left( {\pi - x} \right)} \right]} dx$

$ = \int\limits_0^\pi {\left[ { - \cot x} \right]} dx$

Adding we have

$2I = \int\limits_0^\pi {\left\{ {\left[ {\cot x} \right] + \left[ { - \cot x} \right]} \right\}} dx$

$2I = \int\limits_0^\pi {\left( { - 1} \right)} dx = - \pi $

$\therefore I = - \pi /2$

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