Let $I = \int\limits_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x} dx$ Dividing numerator and denominator by $\cos^4 x$ $I = \int\limits_{0}^{\frac{\pi}{4}} \frac{\tan x \sec^2 x}{1 + \tan^4 x} dx$ Put $\tan^2 x = t$ $\Rightarrow 2 \tan x \sec^2 x \, dx = dt$When $x = 0,\ t = 0$ and $x = \frac{\pi}{4},\ t = 1$$\therefore I = \frac{1}{2} \int\limits_{0}^{1} \frac{dt}{1 + t^2}$ $= \left[ \frac{1}{2} \tan^{-1} t \right]_{0}^{1}$ $= \frac{1}{2} \times \frac{\pi}{4} = \frac{\pi}{8}$