deletions | additions       diff --git a/untitled.tex b/untitled.tex  index df98f8a..c76125e 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
s_1 = \frac{-2}{\sqrt{\pi}}\int_{-\infty}^{\infty} x^2e^{-x^2}\mathrm{ln}(\frac{2}{\sqrt{\pi}}x^2e^{-x^2})\;dx = \gamma + \mathrm{ln}(2) + \frac{1}{2}(\mathrm{ln}(\pi)-1)  \end{equation}  where $\gamma$ is the Euler-Mascheroni constant.  \section{2^{nd} Order Integrals}  When solving for the next integrals we have $H_2(x)=4x^2-2$, and additional complexity. We find by laking a log expansion some progress can be made to finding a closed form for the total entropy of the density state.  We have \begin{equation}  -\int_{-\infty}^{\infty}\frac{e^{-x^2}(16x^4-16x^2+4)}{8\sqrt{\pi}}\left(\sum_{n=1}^\infty \frac{1}{n}\left(\frac{e^{-x^2}(16x^4-16x^2+4)}{8\sqrt{\pi}} -1 \right)^n \right)}{}  \end{equation}