TITLE: $I=\int_{0}^{\pi }\frac{\sin(2x)}{\sin^{4}(x)+\cos^{4}(x)}dx$
QUESTION [5 upvotes]: $$I=\int_{0}^{\pi }\frac{\sin(2x)}{\sin^{4}(x)+\cos^{4}(x)}dx$$
My try:
I took $x=\pi-t$. I got $I=-I$ so $I=0$. Is this correct?

REPLY [4 votes]: Good work!  I'll offer another approach just for the heck of it.  We can see that $\sin^4x$ and $\cos^4 x$ are both symmetric about $\pi/2$.  However, $\sin 2x$ is anti-symmetric about $\pi/2$.  Therefore the entire integrand is anti-symmetric about $\pi/2$.  Since $\pi/2$ is the mid-point of the integration region, the integral must be zero.