_ ^ dx x + x = - Vidyadip

MCQ

$\int_{}^{} {\frac{{dx}}{{\sin x + \cos x}}} = $

A
$\log \tan \left( {\frac{\pi }{8} + \frac{x}{2}} \right) + c$
B
$\log \tan \left( {\frac{\pi }{8} - \frac{x}{2}} \right) + c$
✓
$\frac{1}{{\sqrt 2 }}\log \tan \left( {\frac{\pi }{8} + \frac{x}{2}} \right) + c$
D
None of these

✓

Answer

Correct option: C.

$\frac{1}{{\sqrt 2 }}\log \tan \left( {\frac{\pi }{8} + \frac{x}{2}} \right) + c$

c
(c)$\int_{}^{} {\frac{{dx}}{{\sin x + \cos x}}} = \frac{1}{{\sqrt 2 }}\int_{}^{} {\frac{{dx}}{{\sin x\cos \frac{\pi }{4} + \cos x\sin \frac{\pi }{4}}}} $
$ = \frac{1}{{\sqrt 2 }}\int_{}^{} {{\rm{cosec }}\left( {x + \frac{\pi }{4}} \right)\,dx = \frac{1}{{\sqrt 2 }}\log \tan \left( {\frac{\pi }{8} + \frac{x}{2}} \right)} + c.$

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