Evaluate $\int \sqrt{1 + \sin x} dx$.

$2 (\sin(x/2) - \cos(x/2)) + C$

The correct answer is Option (2) → $2 (\sin(x/2) - \cos(x/2)) + C$

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

$= \int \sqrt{\left(\sin^2 \frac{x}{2} + \cos^2 \frac{x}{2} + 2\sin \frac{x}{2} \cos \frac{x}{2}\right)} dx \quad [∵\sin^2 \frac{x}{2} + \cos^2 \frac{x}{2} = 1]$

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

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