\documentclass[openany,9pt]{book}
\makeatletter\@addtoreset{footnote}{page}\makeatother
\usepackage{mathrsfs}
\usepackage{perpage}
%\Makeperpage[1]{footnote}
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\usepackage{makeidx}
\makeindex
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%here comes fancy definitions%%%%%%%%%%%%%%
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% with this we ensure that the chapter and section
% headings are in lowercase.
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\renewcommand{\thesection}{\thechapter.\arabic{section}}
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\newtheorem{theorem}{قضیه}
\newtheorem{lemma}[theorem]{لم}
\newtheorem{proposition}[theorem]{گزاره}
\newtheorem{corollary}[theorem]{نتیجه}

\theoremstyle{definition}
\newtheorem*{definition}{تعریف}
\newtheorem{remark}{ملاحظه}
\newtheorem*{example}{مثال}
\renewcommand\proofname{برهان}

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\newcommand{\To}{\longrightarrow}
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\numberwithin{figure}{chapter}
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\numberwithin{table}{subsection}
\onehalfspace

\begin{document}
\title{\textbf{}}
\author{\textbf{}\\
\textbf{ }}
\date{}
\maketitle
%\newpage
\tableofcontents
%\listoffigures
%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{ مقدمه}
%\begin{equation*}
\section{انتگرال}
\begin{align*}
\int T_n (x)dx &=\begin{cases}
T_1 (x)& \quad n = 0\\ \\
\frac{1}{4} \Big(T_2 (x)+ T_0 (x)\Big)& \quad n=1\\ \\
\dfrac{T_{n+1} (x)}{n+1}- \dfrac{T_{n-1} (x)}{n-1} & \quad n\geq 2
\end{cases}\\ \\
\int U_n (x)dx&=\begin{cases}
T_1 (x)& \,\,\qquad  \quad n = 0\\ \\
\frac{1}{2} \Big(T_2 (x)+ T_0 (x)\Big)& \,\, \qquad  \quad n=1\\ \\
\dfrac{T_{n+1} (x)}{n+1} & \,\, \qquad  \quad n\geq 2
\end{cases}
\end{align*}
%\end{equation*}

\indexincontents
\printindex
\end{document}