# Definite integral $\int_0^{\pi/2} \frac{\sin^{1/3}(x) \, \mathrm{d}x}{\sin^{1/3}(x) + \cos^{1/3}(x)}$

I am struggling with a particular definite integral. Here it goes:

$$\int_0^{\pi/2} \frac{\sin^{1/3}(x) \, \mathrm{d}x}{\sin^{1/3}(x) + \cos^{1/3}(x)}$$

I have tried several substitutions but haven't succeeded. Mathematica says that the result is $\pi/4$, and it even gives an analytic solution for the indefinite integral. However, it doesn't really help me to see how it is worked out.

Would you please have a look and hint me in the direction/solve it? Thanks a lot.

SSF

It is just $\frac{\pi}{4}$. Apply the substitution $x\mapsto\frac{\pi}{2}-x$ and add the resulting integrals to get $\int_{0}^{\pi/2}1\,dx.$