\documentclass[12pt]{amsart}

\usepackage{amsmath, amssymb, amsthm}
\usepackage[margin=0.5in]{geometry}
\usepackage[pdflang={en-US}]{hyperref}
\usepackage[shortlabels]{enumitem}

\usepackage{tabu}

\renewcommand{\thesection}{\Alph{section}}

\newcommand{\scs}[1]{\begin{scshape}#1\end{scshape}}
\newcommand*{\inv}{^{-1}}
\newcommand*{\RR}{\mathbb{R}}
\newcommand*{\QQ}{\mathbb{Q}}
\newcommand*{\ZZ}{\mathbb{Z}}
\newcommand*{\NN}{\mathbb{N}}

\newcommand*{\inte}[2][x]{\int #2\, d#1}
\newcommand*{\der}[2][x]{\frac{d}{d#1}\!\left[ #2 \right]}

\newtheorem*{theorem}{Theorem}
\theoremstyle{definition}
\newtheorem*{definition}{Definition}
\newtheorem{problem}{}
\newtheorem*{challenge}{Challenge}
\newtheorem*{example}{Example}
\newtheorem*{solution}{Solution}

\newcommand{\thm}[1]{\begin{theorem}#1\end{theorem}}
\newcommand{\prob}[1]{\begin{problem}#1\end{problem}}
\newcommand{\dfn}[1]{\begin{definition}#1\end{definition}}
\newcommand{\chal}[1]{\begin{challenge}#1\end{challenge}}
\newcommand{\eg}[1]{\begin{example}#1\end{example}}
\newcommand{\sol}[1]{\begin{solution}#1\end{solution}}

\newcommand{\abs}[1]{\left| #1 \right|}

\title{Methods for trig integrals}
\date{September 25, 2020}

\begin{document}

\ \vspace{-.25in}

\maketitle

\vspace{-.15in}


\thm{[Pythagorean identities]
\begin{align*}  \sin^2 x + \cos^2 x &= 1 \\
\tan^2 x + 1 &= \sec^2 x \\
1 + \cot^2 x &= \csc^2 x 
\end{align*} 
}

\thm{[Half-angle formulas]
\begin{align*}
 \sin^2 x &= \frac{ 1- \cos 2x}{2} \\ \cos^2 x &= \frac{ 1 + \cos 2x}{2}
 \end{align*}
}

\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabu} to .85\textwidth {|X[l]  X[l]|}
\hline
$\displaystyle \int\sin^m x \cos^n x\ dx$ & \textbf{Strategy} \\
%\hline
If $n$ is odd and positive \dots & \dots keep $\cos x\ dx$, rewrite everything else in terms of $\sin x$, then sub in $u = \sin x$ (with $ du = \cos x\ dx$) \\
If $m$ is odd and positive \dots & \dots keep $\sin x\ dx$, rewrite everything else in terms of $\cos x$, then sub in $u = \cos x$ (with $ (-1)du = \sin x\ dx$) \\
If $m$ and $n$ are both even non-negative integers \dots & \dots use half-angle formulas to convert to $\cos 2x$ and refer back to this table \\
\hline
\end{tabu}
\end{center}

\vspace{.2in}

\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabu} to .85\textwidth {|X[l]  X[l]|}
\hline
$\displaystyle \int \tan^m x \sec^n x\ dx$ & \textbf{Strategy} \\
%\hline
If $n$ is even \dots & \dots keep $\sec^2 x\ dx$, rewrite everything else in terms of $\tan x$, then sub in $u = \tan x$ (with $ du = \sec^2 x\ dx$) \\
If $m$ is odd \dots & \dots keep $\tan x \sec x\ dx$, rewrite everything else in terms of $\sec x$, then sub in $u = \sec x$ (with $ du = \tan x \sec x\ dx$) \\
If $m$ is even and $n$ is odd \dots & \dots rewrite everything in terms of $\sec x$ and apply the reduction formula below \\
If you want \dots & \dots convert to $\sin x$ and $\cos x$ and see the other table \\
\hline
\end{tabu}
\end{center}

\thm{[Reduction formula]
$\displaystyle \int sec^n x\ dx = \frac{\sec^{n-2}x \tan x}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2}x\ dx$ (provided $n \neq 1$).
}

\end{document}