For $x∈(0,\frac{\pi}{2}),\int\frac{\sin x + \cos x}{\sqrt{\sin 2x}}dx$ is equal to

$\sin^{-1}(\sin x-\cos x) + C$, C is an arbitary constant

The correct answer is Option (3) → $\sin^{-1}(\sin x-\cos x) + C$, C is an arbitary constant

Given integral:

$\int \frac{\sin x + \cos x}{\sqrt{\sin 2x}}\,dx$

Let $I = \int \frac{\sin x + \cos x}{\sqrt{\sin 2x}} dx$

Put $u = \sin x - \cos x \Rightarrow du = (\cos x + \sin x)\,dx$

Also, $\sin 2x = 2 \sin x \cos x = 1 - (\sin x - \cos x)^2$

Hence after substitution and simplification, the integral becomes:

$I = \sin^{-1} (\sin x - \cos x) + C$

Final Answer:

${ \sin^{-1} (\sin x - \cos x) + C }$