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% AUTHOR: Amlal El Mahrouss
% PURPOSE: Lemmas for Integrals.
\documentclass[11pt, a4paper]{article}
\usepackage{graphicx}
\usepackage{listings}
\usepackage{xcolor}
\usepackage{hyperref}
\usepackage{mathtools}
\usepackage{amsfonts}
\usepackage[margin=0.5in,top=1in,bottom=1in]{geometry}
\title{Lemmas for Integrals.}
\author{Amlal El Mahrouss\\amlal@nekernel.org}
\date{February 2026}
\begin{document}
\maketitle
\begin{center}
\rule[0.01cm]{17cm}{0.01cm}
\end{center}
\section{Definitions}
We assume that $u \in \mathbb{R}, \quad \forall u \in \mathbb{R} : u < \pi, \quad 0 < u$
\begin{equation}
I_{1} \coloneqq \int_{u_{0}}^{u} S(t) dt, \quad t = u, \quad u > 0
\end{equation}
Where $u, u_{0} > 0, \quad u, u_{0} \in \mathbb{R}$, such that $S : (u, \pi) \to \mathbb{R}$.
\section{Lemmas}
\end{document}
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