_ ^ 1 + x 2 ;dx = - Vidyadip

MCQ

$\int_{}^{} {\sqrt {1 + \sin \frac{x}{2}} \;dx = } $

A
$\frac{1}{4}\left( {\cos \frac{x}{4} - \sin \frac{x}{4}} \right) + c$
B
$4\left( {\cos \frac{x}{4} - \sin \frac{x}{4}} \right) + c$
✓
$4\left( {\sin \frac{x}{4} - \cos \frac{x}{4}} \right) + c$
D
$4\left( {\sin \frac{x}{4} + \cos \frac{x}{4}} \right) + c$

✓

Answer

Correct option: C.

$4\left( {\sin \frac{x}{4} - \cos \frac{x}{4}} \right) + c$

c
(c)$\int_{}^{} {\sqrt {1 + \sin \frac{x}{2}\,} } dx = \int_{}^{} {\sqrt {\left( {{{\sin }^2}\frac{x}{4} + {{\cos }^2}\frac{x}{4} + 2\sin \frac{x}{4}\cos \frac{x}{4}} \right)} \,dx} $
$ = \int_{}^{} {\left( {\sin \frac{x}{4} + \cos \frac{x}{4}} \right)\,dx = 4\left( {\sin \frac{x}{4} - \cos \frac{x}{4}} \right)} + c$.

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