Let $\left(2^{1-\mathrm{a}}+2^{1+\mathrm{a}}\right), f(\mathrm{a}),\left(3^{\mathrm{a}}+3^{-\mathrm{a}}\right)$ be in A.P. and $\alpha$ be the minimum value of $f(\mathrm{a})$. Then the value of the integral $\int\limits_{\log _e(\alpha-1)}^{\log _e(\alpha)} \frac{d x}{\left(e^{2 x}-e^{-2 x}\right)}$ is :

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

If $\alpha=1$ and $\beta=1+i \sqrt{2}$, where $i=\sqrt{-1}$ are two roots of the equation

$x^3+a x^2+b x+c=0, a, b, \in \mathbb{R}$, then $\int_{-1}^1\left(x^3+a x^2+b x+c\right) d x$ is equal to: