Definite Integration

Let \(f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbf{R}\) be a differentiable function such that \(f(0)=\frac{1}{2}\). If the \(\lim _\limits{x \rightarrow 0} \frac{x \int_0^x f(\mathrm{t}) \mathrm{dt}}{\mathrm{e}^{x^2}-1}=\alpha\), then \(8 \alpha^2\) is equal to :