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