Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$. Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, \mathrm dx.$$
How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead anywhere (I think).
Any guidelines please?

Symmetry! This is the same as the integral with $\cos^3 x$ on top. 
If that is not obvious from the geometry, make the change of variable $u=\pi/2-x$. 
Add them, you get the integral of $1$. So our integral is $\pi/4$.
