Gaussian Distribution (Probability Function Approximations)

I took the derivative of a Gaussian distribution, $$\frac{\mathrm{d} }{\mathrm{d} x}\frac{1}{\sqrt{2\pi \sigma^2}}e^{-(x-\mu)^2/{2\sigma^2}}=-\frac{e^{-\frac{(x-\mu )^2}{2 \sigma ^2}} (x-\mu )}{\sqrt{2 \pi } \sigma ^2 \sqrt{\sigma ^2}} $$, in order to find the type of terms I am looking - looking for something like $$x e^{x^2}$$ I expect $$\mu$$ to be zero because I am centering the distribution around zero, and the width of the distribution ($$\sigma$$) to depend on the linearly on n, the depth.

The first approximation I tried was using a simple a simple exponential; $$x!\approx e^{x}$$.  The approximation was not that good, but it was simple.  This did not pan as terms canceled in the integral. The next approximation I tried was $$x!\approx c^{x^2}$$.   The approximation still rose too quickly, and then integrating yielded the error function - not good.

The final approximation that I am still working on is the Sterling Approximation.  Now I am just trying a power expansion, because I cannot integrate $$\frac{n^{\frac{1}{2}+n} p^{-1+n} \int (n-r)^{-\frac{1}{2}-n+r} r^{-\frac{1}{2}-r} \, dr}{\sqrt{2 \pi }}$$.