Let \( y = f(x) \) be the solution of the differential equation \(\displaystyle \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}},\ -1 < x < 1 \) such that \( f(0) = 0 \). If \(\displaystyle 6 \int_{-1/2}^{1/2} f(x)\, dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to __________.
