Correct answer: 15
\(\left|\frac{120}{\pi^3} \int_0^\pi \frac{x^2 \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x\right|\)
\( \Rightarrow\left|\frac{120}{\pi^3} \int_0^{\pi / 2} \frac{\sin x \cos x}{\sin ^4 x+\cos ^4 x}\left(x^2-(\pi-x)^2\right) d x\right|\)
\(\Rightarrow\left|\frac{120}{\pi^3} \int_0^{\pi / 2} \frac{\sin x \cdot \cos x}{\sin ^4 x+\cos ^4 x}\left(2 \pi x-\pi^2\right) d x\right| \)
\(\Rightarrow\left|\frac{120}{\pi^3}\left[2 \pi \int_0^{\pi / 2} \frac{x \sin x \cdot \cos x}{\sin ^4 x+\cos ^4 x} d x-\pi^2 \int_0^{\pi / 2} \frac{\sin x \cdot \cos x}{\sin ^4 x+\cos ^4 x} d x\right]\right|\)
\(\Rightarrow\left|\frac{120}{\pi^3}\left[2 \pi \times \frac{\pi^{\pi / 2}}{4} \int_0^{\pi / 2} \frac{\sin x \cdot \cos x}{\sin ^4 x+\cos 4} d x-\pi^2 \int_0^{\pi / 2} \frac{\sin x \cdot \cos x}{\sin ^4 x+\cos ^4 x} d x\right]\right| \)
\(\Rightarrow\left|\frac{120}{\pi^3}\left[\frac{-\pi^2}{2} \int_0^{\pi / 2} \frac{\sin x \cdot \cos x}{\sin ^4 x+\cos ^4 x} d x\right]\right| \)
\(\Rightarrow \left\lvert\, \frac{120}{\pi^3}\left[\frac{-\pi^2}{2} \int_0^{\pi / 2} \frac{\sin x \cdot \cos x}{1-2 \sin ^2 x \cos { }^2 x} d x\right]\right|\)
\(\Rightarrow \left\lvert\, \frac{120}{\pi^3}\left[\frac{-\pi^2}{2} \int_0^{\pi / 2} \frac{\frac{1}{2} \sin ^2 x}{1-\frac{1}{2} \sin ^2 2 x} d x\right]\right|\)
\(\Rightarrow \left\lvert\, \frac{120}{\pi^3}\left[\frac{-\pi^2}{2} \int_0^{\pi / 2} \frac{\sin 2 x}{2-\sin ^2 2 x} d x\right]\right|\)
\(\Rightarrow\left|\frac{120}{\pi^3}\left[\frac{-\pi^2}{2} \int_0^{\pi / 2} \frac{\sin 2 \mathrm{x}}{1+\cos ^2 2 \mathrm{x}} \mathrm{dx}\right]\right|\)
take \(\cos 2 \mathrm{x}=\mathrm{t},-2 \sin 2 \mathrm{xdx}=\mathrm{dt}\)
\(\Rightarrow\left|\frac{120}{\pi^3}\left[\frac{-\pi^2}{2} \int_1^{-1} \frac{-\frac{1}{2} \mathrm{dt}}{1+\mathrm{t}^2}\right]\right|\)
\(\Rightarrow \left\lvert\, \frac{120}{\pi^3}\left[\frac{-\pi^2}{2} \int_{-1}^1 \frac{\frac{1}{2} \mathrm{dt}}{1+\mathrm{t}^2}\right]\right|\)
\(\Rightarrow \left\lvert\, \frac{120}{\pi^3} \times\left[\frac{-\pi^2}{4}\left(\tan ^{-1}(\mathrm{t})\right)_{-1}^1\right]\right|\)
\(\Rightarrow\left|\frac{120}{\pi^3} \times\left[\frac{-\pi^2}{4} \times \frac{\pi}{2}\right]\right|\)
\( \Rightarrow 15\)
