\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\usetikzlibrary{patterns}
\begin{document}
\def\a{-.3}
\def\x{2.4}
\def\b{5.7}
\begin{tikzpicture}
\begin{axis}[
axis lines = middle,
xlabel={},
xlabel style={
%yshift=+7pt, 
%xshift=3cm,  % Décale l'étiquette vers la droite (si nécessaire)
},
xtick=\empty,ytick=\empty,
ylabel={},
height=4cm,
width=0.8\linewidth,
enlargelimits=true,
enlarge x limits={abs=3mm}, 
enlarge y limits={abs=1mm}, 
extra x ticks={\a,\x,\b},
extra x tick style={grid=major,
tick label style={
rotate=0,anchor=north east},yshift=0mm},
extra x tick labels={$a$,\color{red} $c$,$b$},
]
\addplot[red, thick, smooth, domain=\a:\x, pattern=north east lines, pattern color=lightgray] {(x^2+1-x^3/8)/5-1}\closedcycle;
\addplot[red, thick, smooth, domain=\x:\b, pattern=north east lines, pattern color=lightgray] {(x^2+1-x^3/8)/5-1}\closedcycle;
\addplot[black, ultra thick, smooth, domain=-1:7] {(x^2+1-x^3/8)/5-1} node[above] {$\mathcal{C}_f$};
%\node[fill=white, inner sep=0pt] at (axis cs:2.5,.35) {\(F(x)\)};
\end{axis}
\end{tikzpicture}
\end{document}