_ ^ 2 + 3x ;. ; 3x ;dx = - Vidyadip

MCQ

$\int_{}^{} {\sqrt {2 + \sin 3x} \;.\;\cos 3x\;dx = } $

A
$\frac{2}{9}{(2 + \sin 3x)^{1/2}} + c$
B
$\frac{2}{3}{(2 + \sin 3x)^{2/3}} + c$
C
$\frac{2}{3}{(2 + \sin 3x)^{3/2}} + c$
✓
$\frac{2}{9}{(2 + \sin 3x)^{3/2}} + c$

✓

Answer

Correct option: D.

$\frac{2}{9}{(2 + \sin 3x)^{3/2}} + c$

d
(d) Put $2 + \sin 3x = t \Rightarrow 3\cos 3x\,dx = dt,$ then
$\int_{}^{} {\sqrt {2 + \sin 3x} \cos 3x\,dx} = \frac{1}{3}\int_{}^{} {{t^{1/2}}dt} = \frac{2}{9}{t^{3/2}} + c$
$ = \frac{2}{9}{(2 + \sin 3x)^{3/2}} + c$.

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