The product of all possible values of $\alpha$, for which

$\lim \limits_{x \rightarrow 0}\left(\frac{1-\cos (\alpha x) \cos ((\alpha+1) x) \cos ((\alpha+2) x)}{\sin ^2((\alpha+1) x)}\right)=2$, is :

The value of the integral $\int\limits_0^{\infty} \frac{\log _e(x)}{x^2+4} d x$ is:

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)}{3}$ for all $x, y \in \mathbb{R}$, and $f^{\prime}(0)=3$. Then the minimum value of the function $g(x)=3+e^x f(x)$, is:

The value of the integral $\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}}\left(\frac{4-\operatorname{cosec}^2 x}{\cos ^4 x}\right) d x$ is :