_0^ 2 x 1+ x 2 d x= - Vidyadip

MCQ

$\int_0^{2 x} \sqrt{1+\sin \frac{x}{2}} d x=$

A
$0$
B
2
✓
8
D
4

✓

Answer

Correct option: C.

8

(C)
$\int_0^{2 \pi} \sqrt{1+\sin \frac{x}{2}} d x=\int_0^{2 \pi} \sqrt{\left(\sin \frac{x}{4}+\cos \frac{x}{4}\right)^2} d x$
$=\int_0^{2 \pi}\left(\sin \frac{x}{4}+\cos \frac{x}{4}\right) d x$
$\ldots .\left[\because x \in(0,2 \pi),\left(\sin \frac{x}{4}+\cos \frac{x}{4}\right)>0\right]$
$=4\left[-\cos \frac{x}{4}+\sin \frac{x}{4}\right]_0^{2 \pi}$
$\begin{array}{l}=4(0+1+1-0) \\ =8\end{array}$

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