Let y = f ( x ) be a thrice differentiable function in (− 4, 4). Let the tangents to the curve y = f (x) at (2, f(2)) and (3, f(3)) make angles π/6 and π/3, respectively with positive x-axis. If \(54 \int\limits_{1}^{3}\left(2-(f'(t))^{2}\right) f''(t) d t = \alpha\sqrt{3}\) where α is an integers, then the value of α equals