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%%
%% Ein Beispiel der DANTE-Edition
%%
%% 2. Auflage
%%
%% Beispiel 09-08-44 auf Seite 469.
%%
%% Copyright (C) 2016 Herbert Voss
%%
%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
%%
%% See http://www.latex-project.org/lppl.txt for details.
%%
%%
%% ====
% Show page(s) 1
%%
%%
\documentclass[]{exaarticle}
\pagestyle{empty}
\setlength\textwidth{352.81416pt}
\AtBeginDocument{\setlength\parindent{0pt}}
%StartShownPreambleCommands
\usepackage{mathtools}
\newcommand*\diff{\mathop{}\!\mathrm{d}}
%StopShownPreambleCommands
\begin{document}
\begin{align*}
\MoveEqLeft[6]
\frac{A}{x-x_1},\ \frac{A}{\left(x-x_1\right)^k},\ \frac{Ax+b}{x^2+px+q},\
\frac{Ax+b}{\left(x^2+px+q\right)^k} \text{ mit } p^2<4q \text{ und } A\ne 0\\
\int\frac{A}{x-x_1}\diff x &= A\ln\left|x-x_1\right|+C\\
\int\frac{A}{\left(x-x_1\right)^k}\diff x
&=-\frac{A}{(k-1)\left(x-x_1\right)^{k-1}}+C\\
\int\frac{\diff x}{x^2+px+q}
&= \frac{2}{\sqrt{4q-p^2}}\cdot\arctan\frac{2x+p}{\sqrt{4q-p^2}}+C\\
\int\frac{Ax+B}{x^2+px+q}\diff x
&= \frac{A}{2}\ln\left|x^2+px+q\right| +\frac{2B-Ap}{\sqrt{4q-p^2}}
\cdot\arctan\frac{2x+p}{\sqrt{4q-p^2}}+C
\end{align*}
\end{document}
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