The value of the integral $\int\limits_{-1}^1\left(\frac{x^3+|x|+1}{x^2+2|x|+1}\right) \mathrm{d} x$ is equal to:

The value of the integral $\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\left(\frac{32 \cos ^4 x}{1+e^{\sin x}}\right) d x$ is :

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: