_0^ 2 1 + x 2 ,dx = - Vidyadip

Question

$\int_0^{2\pi } {\sqrt {1 + \sin \frac{x}{2}} \,dx = } $

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Answer

c
(c) $\int_0^{2\pi } {\sqrt {1 + \sin \frac{x}{2}} dx} $

$= \int_0^{2\pi } {\left| {\sin \frac{x}{4} + \cos \frac{x}{4}} \right|dx = 4\left[ {\sin \frac{x}{4} - \cos \frac{x}{4}} \right]} _0^{2\pi }$

$ = 4[1 - 0 - 0 + 1] = 8$.

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