Integrate $\int_0^{\pi/2}\frac{\sin^3t}{\sin^3t+\cos^3t}dt$?

Question

How to integrate

$$ \int_0^{\pi/2}\frac{\sin^3t}{\sin^3t+\cos^3t}dt\,? $$

I tried to use $\sin^3tdt=-(1-\cos^2t)d\cos t$. But the term $\sin^3t$ in the denominator can not be simplified. Can anyone give me a hand? Thanks.

http://math.stackexchange.com/questions/439851/evaluate-the-integral-int-frac-pi2-0-frac-sin3x-sin3x-cos3xdx — lab bhattacharjee, Oct 18 '13 at 08:44

score 6 · Accepted Answer · answered Oct 18 '13 at 08:42

6

General Hint:

By setting $x=\pi/2-y$ you can see that $$\int_0^{\pi/2}\frac{\sin^m(x)}{\sin^m(x)+\cos^m(x)}dx=\pi/4,~~m\in\mathbb R$$

answered Oct 18 '13 at 08:42

Mikasa

1 Answers1