\(\int_{0}^{\frac{\pi}{2}}\frac{\sin^{4}x}{\sin^{4}x+\cos^{4}x} dx \) is equal to

\(\frac{\pi}{4}\)

\(I=\int\limits_{0}^{\frac{\pi}{2}}\frac{\sin^{4}x}{\sin^{4}x+\cos^{4}x} dx \)  ...(1)

$I=\int\limits_{0}^{\frac{\pi}{2}}\frac{\sin^4(\frac{\pi}{2}-x)}{\sin^4(\frac{\pi}{2}-x)+\cos^4(\frac{\pi}{2}-x)}dx=\int\limits_{0}^{\frac{\pi}{2}}\frac{\cos^{4}x}{\sin^{4}x+\cos^{4}x} dx$   ...(2)

Eq. (1) + Eq. (2)

$2I=\int\limits_{0}^{\frac{\pi}{2}}1dx⇒I=\frac{\pi}{4}$