Let f: [- (π)/(2), (π)/(2)] → R be a differentiable function such...

Let, $y = \lim_{x \to 0}\dfrac{x{∫}_{0}^{x}f ( t ) dt}{{e}^{{x}^{2}} - 1}$

$⇒ y = \lim_{x \to 0}\dfrac{x{∫}_{0}^{x}f ( t ) dt}{\left(\dfrac{{e}^{{x}^{2}} - 1}{{x}^{2}}\right) \times {x}^{2}}$

We know that, $\lim_{x \to 0}\dfrac{{e}^{{x}^{2}} - 1}{{x}^{2}} = 1$

$⇒ y = \lim_{x \to 0}\dfrac{{∫}_{0}^{x}f ( t ) dt}{x}$

Applying L-hospital's rule and Newton Leibnitz Theorem we get,

$⇒ y = \lim_{x \to 0}\dfrac{f ( x )}{1}$

$⇒ y = f\left(0\right)$

It is given that, $f ( 0 ) = \dfrac{1}{2}$

$⇒ y = \dfrac{1}{2}$

$⇒ \alpha = \dfrac{1}{2}$

$⇒ 8{\alpha}^{2} = 2$