How to solve this entropy integral?
The integration over a vector $\mathbf x$ as given by you means something like

$$\int f(\mathbf x) d\mathbf x=\int\dots\int f(x_1,\dots,x_n)dx_1\dots dx_n$$

So, you integrate over each component separately. Now, to solve your integral, you should first split the $\log$ into a difference and evaluate both integrals separately (exploiting the fact that $\exp(ab)=\exp(a)\exp(b)$. Then you will get a lot of similarly structured integrals that depend only on $\sigma_i$. If you solve them, you'll get the solution.