Calculating with Mathematica, one can have $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}\,\mathrm dt=\frac{\pi}{4}.$
- How can I get this formula by hand? Is there any simpler idea than using $u = \sin t$?
- Is there a simple way to calculate $ \int_0^{\pi/2}\frac{\sin^n t}{\sin^n t+\cos^n t}\,\mathrm dt $ for $n>3$?
- Could anyone come up with a reference for this exercise?