<s>[INST] Evaluate $\int_0^{π/2} \frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}\,dx$ Find the value of
$$\int_0^{\pi/2} \frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}\,dx$$
I'm having a very tough time solving this question, can any body give me some hints?
 [/INST] Note that after lettimg $x=\pi/2-t$, we obtain
$$I:=\int_{0}^{\frac{\pi}{2}} \frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}\,dx
=\int_{0}^{\frac{\pi}{2}} \frac{(\frac{\pi}{2}-t)\sin t\cos t}{\sin^4 t+\cos^4 t}\,dt=\int_{0}^{\frac{\pi}{2}} \frac{\frac{\pi}{2}\sin t\cos t}{\sin^4 t+\cos^4 t}\,dt-I$$
which implies that
\begin{align*}
I&=\frac{\pi}{4}\int_{0}^{\frac{\pi}{2}} \frac{\sin t\cos t}{\sin^4 t+\cos^4 t}\,dt\quad (s:=\sin(t))\\
&=\frac{\pi}{4}\int_{0}^{1}\frac{s}{s^4+(1-s^2)^2}\,ds
\\
&=\frac{\pi}{8}\int_{0}^{1}\frac{4s}{(2s^2-1)^2+1}\,ds
\\
&=\frac{\pi}{8}\left[\arctan(2s^2-1)\right]_0^1=\frac{\pi^2}{16}.
\end{align*}
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