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