A Calculus problem in GRE math sub test

Question

$$\int_0^{\frac{\pi}{2}} \frac{\sin^3(t)}{\sin^3(t)+\cos^3(t)}$$

I tried to use $x=\tan(t)$ to transform but it's really a mess..

Answer 1

Hint: Use symmetry about $\frac{\pi}{4}$ (or equivalently the substitution $t=\frac{\pi}{2}-u$) to show that your integral is the same as $\int_0^{\pi/2}\frac{\cos^3 t}{\sin^3 t+\cos^3 t}\,dt$.

But the sum of the two integrals is $\frac{\pi}{2}$.