$\int_0^{\pi/2} \frac{\sqrt{\sin(x)}}{\sqrt{\sin(x)}+\sqrt{\cos(x)}} dx,$ with $x=\frac{\pi}{2}-t$

Question

I already solved a few integrals with substitution but in this case I have no idea how to start. How to solve the integral $$\int_0^{\pi/2} \frac{\sqrt{\sin(x)}}{\sqrt{\sin(x)}+\sqrt{\cos(x)}} dx,$$where $x=\frac{\pi}{2}-t$ with substitution, can you tell me how to start? It would be great!

Answer 1

$$\int_a^b f(x) dx=\int_a^b f(a+b-x) dx$$ $$\tag1I=\int_0^{\pi/2} \frac{\sqrt{\sin(x)}}{\sqrt{\sin(x)}+\sqrt{\cos(x)}} dx,$$ Replace $x$ by $\frac{\pi}2-x$. $$I=\int_0^{\pi/2} \frac{\sqrt{\sin(\frac{\pi}2-x)}}{\sqrt{\sin(\frac{\pi}2-x)}+\sqrt{\cos(\frac{\pi}2-x)}} dx,$$ $$\tag2I=\int_0^{\pi/2} \frac{\sqrt{\cos(x)}}{\sqrt{\cos(x)}+\sqrt{\sin(x)}} dx,$$ Adding $(1)$ and $(2)$, $$2I=\int_0^{\pi/2}dx$$ which can be solved easily.

Answer 2

after the substitution we have the integral $$-\int_{\pi/2}^0{\frac {\sqrt {\cos \left( t \right) }}{\sqrt {\cos \left( t \right) } +\sqrt {\sin \left( t \right) }}}dt$$