What is the integration of $\int {\sqrt {1 + \sin x} } dx$ ~~~~~[NCERT Exemp. Q. 9,Page 164 ]

Let $I = \int {\sqrt {1 + \sin x} } dx$


$ = \int {\sqrt {{{\left( {\sin \frac{x}{2} + \cos \frac{x}{2}} \right)}^2}} } dx = \int {\left( {\sin \frac{x}{2} + \cos \frac{x}{2}} \right)} dx$


$ = – \cos \frac{x}{2} \cdot 2 + \sin \frac{x}{2} \cdot 2 + C = – 2\cos \frac{x}{2} + 2\sin \frac{x}{2} + C$

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