\displaystyle \int \limits_0^{2 \pi} \frac{dx}{\sin^4 x + \cos^4 x} = \int \limits_0^{2 \pi} \frac{\frac{1}{\cos^2 x}}{\tan^4 x + 1} \cdot \frac{dx}{\cos^2 x} = \int \limits_0^{2 \pi} \frac{\tan^2 x + 1}{\tan^4 x + 1} \cdot \frac{dx}{\cos^2 x}