_0^ /2 x + x 1 + x ,dx = - Vidyadip

MCQ

$\int_0^{\pi /2} {\frac{{x + \sin x}}{{1 + \cos x}}\,dx = } $

A
$ - \log 2$
B
$\log 2$
✓
$\frac{\pi }{2}$
D
$0$

✓

Answer

Correct option: C.

$\frac{\pi }{2}$

c
(c) $\int_0^{\pi /2} {\frac{{x + \sin x}}{{1 + \cos x}}dx = \int_0^{\pi /2} {\frac{{x + \sin x}}{{2{{\cos }^2}\frac{x}{2}}}dx} } $

$ = \frac{1}{2}\int_0^{\pi /2} {x{{\sec }^2}\frac{x}{2}} dx + \int_0^{\pi /2} {\tan \frac{x}{2}dx} $.

$ = \left| {\,x\tan \frac{x}{2}\,} \right|_0^{\pi /2} = \frac{\pi }{2}\tan \frac{\pi }{4} = \frac{\pi }{2}$.

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