Evaluate $\int_0^1 \frac{1}{\sqrt x+\sqrt {(1-x)}}dx$ Evaluate $I=\int_0^1 \frac{1}{\sqrt x+\sqrt {(1-x)}}dx$. I applied $x=\sin^2\theta$,that makes $I=\int_0^{\pi/2} \frac{\sin2\theta}{\sin\theta+\cos\theta}d\theta$...--prophetes.ai

Evaluate $\int_0^1 \frac{1}{\sqrt x+\sqrt {(1-x)}}dx$ Evaluate $I=\int_0^1 \frac{1}{\sqrt x+\sqrt {(1-x)}}dx$. I applied $x=\sin^2\theta$,that makes $I=\int_0^{\pi/2} \frac{\sin2\theta}{\sin\theta+\cos\theta}d\theta$,but the further proceedings makes $I$ quite tedious. I need to know is there some elegant transformation which can simplify the calculations. Any suggestions are heartily welcome

Hint:

$$\sin2\theta=(\sin\theta+\cos\theta)^2-1$$

and $\sin\theta+\cos\theta=\sqrt2\sin\left(\dfrac\pi4+\theta\right)$

Use Integral of $\csc(x)$