\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage[margin=1in]{geometry}

\begin{document}

\title{PHYS 2380 - Assignment 4 - Integral table}
\author{Useful integrals for Assignment 4}
\maketitle

\bigbreak
The following integrals will simplify some problems in Assignment $4$. Rather than look them up in a table or calculate them numerically, it may be easier to gather them together. We have used some of them in class already.

\bigbreak 

\begin{equation}
\int_{-\infty}^{\infty} e^{-u^2} du = \sqrt{\pi}
\end{equation}

\bigbreak 

\begin{equation}
\int_{-\infty}^{\infty} u^2 e^{-u^2} du = \frac{\sqrt{\pi}}{2}
\label{u2}
\end{equation}

\bigbreak 

\begin{equation}
\int_{-\infty}^{\infty} u^4 e^{-u^2} du = \frac{3 \sqrt{\pi}}{4}
\end{equation}

\bigbreak

In addition to these integrals, you may also need to evaluate integral \ref{u2} (above) over a particular limit. This integral can be simplified somewhat using integration by parts, but it is by no means a clean solution. When (if) you find the need to evaluate the integral with limits that are not over all space, just check for the appropriate lower limit $x$ on the table below. The corresponding value of the integral $F(x)$ is given. 
\bigbreak 

Let us define
\begin{equation}
F(x) = \int_{x}^{\infty} u^2 e^{-u^2} du.
\end{equation}
 
\bigbreak
\begin{center}
    \begin{tabular}{| l | l |}
    \hline
    $x$ & $F(x)$ \\ \hline
    $\frac{\sqrt{\pi}}{3}$ & $0.3871$ \\ \hline
    $\frac{\sqrt{\pi}}{2}$ & $0.2951$ \\ \hline
    $1$ & $0.2536$ \\ \hline
    $\sqrt{2}$ & $0.1159$ \\ \hline
    $\sqrt{3}$ & $0.0495$ \\ \hline
    $\sqrt{\pi}$ &  $0.0437$ \\ \hline
    $2$ & $0.0204$ \\ \hline
    $\sqrt{6}$ & $0.0033$ \\ \hline
    $\sqrt{7}$ & $0.0013$ \\ \hline
    $\sqrt{8}$ & $5.025\times 10^{-4}$ \\ \hline
    $3$ & $1.949\times 10^{-4}$ \\ \hline
    $4$ & $2.319\times 10^{-7}$ \\ \hline
    \end{tabular}
\end{center}

\end{document} 