calculus.tex

\documentclass[main]{subfiles}
\begin{document}
\begin{multicols}{2}

\paragraph{Special Integrals}
\begin{gather}
\int^\infty_\infty\dd{x}e^{-ax^{2}+cx} = \sqrt{\frac{\pi}{a}}e^{\frac{c^2}{4a}}\\
\int^\infty_\infty\dd{x}e^{-(x,Ax)+(c,x)} = \frac{\pi^{\frac{N}{2}}}{\sqrt{\det A}} e^{\frac{1}{4}(c,A^{-1}c)}\\
(2\pi)^3\delta^3(\vb{x}) = \int_\infty^\infty e^{i\vb{k}\cdot \vb{x}}\dd{\vb{k}}
\\
\lim_{t\rightarrow \infty}
\frac{\sin^2(\alpha t)}{t\alpha^2} = \pi \delta(\alpha)
\\
\int_0^t e^{i\omega t'} \,dt' = \frac{2\sin(\frac{\omega t}{2})}{\omega}
\end{gather}
\end{multicols}

Integral tricks
\begin{multicols}{2}
    Tangent half-angle substitution $t = \tan \frac{x}{2}$
    \begin{gather*}
        \dd x = \frac{2}{1 + t^2}\dd t,\quad \sin{x} = \frac{2t}{1 + t^2},\quad \cos{x} = \frac{1 - t^2}{1 + t^2}
    \end{gather*}
\end{multicols}

\end{document}