Funcion Delta De Dirac

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{geometry} [a4paper,lmargin=1.5cm,rmargin=1.5cm,Botton=1.5cm,top=1.5cm]
\usepackage{fancyhdr}
\pagestyle{fancy}
\usepackage{lastpage}
\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{color}
\usepackage{amsmath}
\usepackage[T1]{fontenc}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage{cancel}
\usepackage[all]{xy}
\usepackage{pstricks}
\usepackage{pst-all}
\usepackage{pst-solides3d}
\usepackage{fancybox}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usepackage{gnuplottex}
\tikzset{flippedeventlabel/.append style={align=center}} \usetikzlibrary{matrix.skeleton} \usetikzlibrary[shapes,arrows,positioning,fit,backgrounds,intersections,shadows,calc,shadings]
\usetikzlibrary{positioning}
\usetikzlibrary{decorations.text} \usetikzlibrary{decorations.pathmorphing} \pgfdeclarelayer{background layer}
\pgfdeclarelayer{foreground layer}
\pgfsetlayers{background layer,main,foreground layer}
\usepackage{color,colortbl}
\usepackage{lscape}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usetikzlibrary{datavisualization} \usetikzlibrary[shapes,arrows.meta,positioning,fit,backgrounds,intersections,shadows,calc,datavisualization.formats.functions] \usetikzlibrary{patterns} \usepackage[colorlinks=true,linkcolor=black,citecolor=black,filecolor=magenta,urlcolor=blue]{hyperref}
\urlstyle{same}
%Paquete de estilo de referencias
\begin{document}
\pagecolor{blue!55!green!90}
\tikz
{\draw (0,0) node[right,fill=black,text width=7.5cm,rounded corners=10pt]
{
\begin{flushright}
\textcolor{white}{\textbf{Sobre Funciones Generalizadas En Física}}\\
\textcolor{yellow}{Nimrod Rodríguez}\\
\href{SITIO WEB}{\underline{\textcolor{red}{enlace:} \textcolor{green}{Código \LaTeX} }}
\end{flushright}
}}

\renewcommand{\sin}{sen}
\begin{tikzpicture}[scale =1,information text/.style={rounded corners=7pt, fill=blue,inner sep=2ex}]
\draw (0,5)[xshift=2.3cm]  node  [left,text width=10cm, information text,scale=1.3]
{\begin{minipage}{10cm}\bf\color{white}
\begin{center}
\shadowbox{Función Delta de Dirac}\\
\end{center}
 En los primeros ejercicios del libro que se usaba como texto en un curso de Física, aparecían estos problemas de cálculo:\\
Dada la siguiente definición:\\
\tikz \draw (0,0) node[fill=black,text width=9cm]
{
\[
\delta (x) =
\left\{
\begin{array}{rl}
0, & si\  x \neq 0\\
\infty, & si\  x = 0\\
\end{array}
\hspace{0.3cm} con: \hspace{0.3cm} \displaystyle\int_{-\infty}^{+\infty} \!\!\delta (x)\ \mathrm{d}x = 1
\right.
\]
};
Evaluar las siguientes integrales:
\tikz \draw (0,0) node[fill=black,text width=9cm]
{
\begin{itemize}
\item $\displaystyle\int_{-3}^{+1} \!\!(x^3-3x^2+2x-1)\ \delta(x+2)\mathrm{d}x$
\item $\displaystyle\int_{0}^{\infty} \!\![\cos(3x)+2]\ \delta(x-\pi)\mathrm{d}x$
\item $\displaystyle\int_{-1}^{+1} \!\! e^{|x|+3} \ \delta(x-2)\mathrm{d}x$
\end{itemize}
};
Hasta aquí todo iba bien, pero cuando se llegó al ejercicio:\\
Demostrar que:\\
\tikz \draw (0,0)
node[fill=black,text width=9cm]
{
$$\delta (x)\ =\frac{1}{2\pi}\displaystyle\int_{-\infty}^{+\infty} \!\! e^{ikx} \ \mathrm{d}k$$
};
Aquí el asunto era ya de un análisis más cuidadoso.
\end{minipage}};
\end{tikzpicture}
\begin{figure}[ht!]
\centering
\tikz \draw (0,0)
node[fill=black,text width=12cm,inner sep=5ex,rounded corners]
{
\tdplotsetmaincoords{80}{110}
\begin{tikzpicture}[scale=1.5,tdplot_main_coords,radius=0.5pt, information text/.style={rounded corners,color=white, fill=blue,inner sep=2ex}]

\draw[red,thick,<->] (-1,0,0) -- (1,0,0) node[anchor=west]{ };
\draw[orange,thick,<->] (0,-1,0) -- (0,1,0);
\draw[yellow,thick,->] (0,0,0) -- (0,0,3) node[white,anchor=west]{$f_n(x)= \frac{1}{2\epsilon_n}, \ -\epsilon_n \leq x\leq \epsilon_n,\ $0 en otro caso};
\draw (0,1.3,0)node[white]{\textbf{$\epsilon_n$}};
\draw (0,-1.4,0)node[white]{\textbf{$-\ \epsilon_n$}};
\foreach \a in {0.20,0.25,...,0.60}
{
\pgfmathparse{100*\a-10}
\pgfmathsetmacro {\b}{1/(2*\a)};
\pgfmathsetmacro {\c}{-1*\a};
\draw[white] (0,\c,0)--(0,\c,\b)--(0,\a,\b)--(0,\a,0);
}
\draw [xshift=1.85cm] node [right,text width=4cm, information text,scale=0.75]
{
La figura muestra la sugerencia en la que se aproxima la funcion delta de Dirac, mediante funciones rectangulares normalizadas ($A_n=2\epsilon_n\ ×\frac{1}{2\epsilon_n} =1$), y asi lograr
$\displaystyle\int_{-\infty}^{+\infty} \!\!\delta (x)\ \mathrm{d}x = 1$
};
\end{tikzpicture}
};
\end{figure}
\end{document}