A) \[\frac{1}{4}\left( \cos \frac{x}{4}-\sin \frac{x}{4} \right)+c\]
B) \[4\left( \cos \frac{x}{4}-\sin \frac{x}{4} \right)+c\]
C) \[4\left( \sin \frac{x}{4}-\cos \frac{x}{4} \right)+c\]
D) \[4\left( \sin \frac{x}{4}+\cos \frac{x}{4} \right)+c\]
Correct Answer: C
Solution :
\[\int_{{}}^{{}}{\sqrt{1+\sin \frac{x}{2}\,}}dx=\int_{{}}^{{}}{\sqrt{\left( {{\sin }^{2}}\frac{x}{4}+{{\cos }^{2}}\frac{x}{4}+2\sin \frac{x}{4}\cos \frac{x}{4} \right)}\,dx}\] \[=\int_{{}}^{{}}{\left( \sin \frac{x}{4}+\cos \frac{x}{4} \right)\,dx=4\left( \sin \frac{x}{4}-\cos \frac{x}{4} \right)}+c\].
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