_ ^ dx x + 3 x = - Vidyadip

MCQ

$\int_{}^{} {\frac{{dx}}{{\sin x + \sqrt 3 \cos x}}} = $

A
$\log \tan \left( {\frac{x}{2} + \frac{\pi }{2}} \right) + c$
✓
$\frac{1}{2}\log \tan \left( {\frac{x}{2} + \frac{\pi }{6}} \right) + c$
C
$\log \cot \left( {\frac{x}{2} + \frac{\pi }{6}} \right) + c$
D
$\frac{1}{2}\log \cot \left( {\frac{x}{2} + \frac{\pi }{6}} \right) + c$

✓

Answer

Correct option: B.

$\frac{1}{2}\log \tan \left( {\frac{x}{2} + \frac{\pi }{6}} \right) + c$

b
(b)$\int_{}^{} {\frac{{dx}}{{\sin x + \sqrt 3 \cos x}}} = \frac{1}{2}\int_{}^{} {\frac{{dx}}{{\frac{{\sin x}}{2} + \frac{{\sqrt 3 }}{2}\cos x}}} $
$ = \frac{1}{2}\int_{}^{} {\frac{{dx}}{{\sin \left( {x + \frac{\pi }{3}} \right)}}} = \frac{1}{2}\int_{}^{} {{\rm{cosec}}\left( {x + \frac{\pi }{3}} \right)} $
$ = \frac{1}{2}\log \tan \left( {\frac{x}{2} + \frac{\pi }{6}} \right) + c.$

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