_0^ /2 x x ,dx equals - Vidyadip

MCQ

$\int_0^{\pi /2} {x\cot x\,dx} $ equals

A
$ - \frac{\pi }{2}\log 2$
✓
$\frac{\pi }{2}\log 2$
C
$\pi \log 2$
D
$ - \pi \log 2$

✓

Answer

Correct option: B.

$\frac{\pi }{2}\log 2$

b
(b) $I = \int_0^{\pi /2} {x\cot x\,dx} $

Integrating by parts, we get

$[x(\log \sin x)]_0^{\pi /2} - \int_0^{\pi /2} {\log \sin x\,dx} $

$I = - \left( { - \frac{\pi }{2}\log 2} \right) = \frac{\pi }{2}\log 2$.

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