Mathematics Question from JEE Main 2026 · 22 January, Shift 2

Let $[\,]$ be the greatest integer function. If $\alpha=\int_{0}^{64}\left(x^{1/3}-[x^{1/3}]\right)\,dx,$ then $\frac{1}{\pi}\int_{0}^{\alpha\pi}\frac{\sin^2\theta}{\sin^6\theta+\cos^6\theta}\,d\theta$ is equal to _____.