LaTeX Equations from tonight

The following images are LaTeX generated and I made them in the process of doing my homework. They serve here as a reminder of syntax, nostalgia-bait, and examples of the power/shortcomings of LaTeX. Also, check out SAGE.

Equations
Code:

\documentclass[a4paper,10pt]{scrartcl}
\usepackage[utf8x]{inputenc}

%opening
\title{10.8 Feb 4}
\author{Spencer Krum}

\begin{document}

\maketitle

\begin{abstract}

\end{abstract}

\section{Equations}
\begin{enumerate}
\item $v_{mp} = \sqrt{\frac{2k_{b}T}{m}}$
\item $n(v)dv = \frac{4 \pi N}{V}\left ( \frac{m}{2\pik_{b}T} \right )^{3/2}v^{2}e^{\frac{-mv^{2}}{2k_{b}T}}dv$
\item $-4 \, \frac{\pi N k m t x^{3} e^{\left(-\frac{1}{2} \, k m t x^{2}\right)}}{V} + 8 \, \frac{\pi N x e^{\left(-\frac{1}{2} \, k m t x^{2}\right)}}{V}$
\item $-c k m t x^{3} e^{\left(-\frac{1}{2} \, k m t x^{2}\right)} + 2 \, c x e^{\left(-\frac{1}{2} \, k m t x^{2}\right)}$

\end{enumerate}


\end{document}