\documentclass{standalone}
\standaloneconfig{border=2mm 2mm 2mm 2mm}
\usepackage{pgfplots}
\usepackage{amsmath}
\pgfplotsset{compat=newest}

% Define a function for the integrand
\pgfmathdeclarefunction{integrand}{1}{%
  \pgfmathparse{(1)/(sqrt(1+#1^2+ln(#1)^2))}%
}

% Define a function for numerical integration using trapezoidal rule
\pgfmathdeclarefunction{trapz}{3}{%
  \pgfmathparse{(#3-#2)/2 * (integrand(#2) + 2*integrand((#2+#3)/2) + integrand(#3))}%
}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
    xlabel=$x$,
    ylabel={$f(x)$},
    xmin=0, xmax=20, % Adjust x-axis limits
    ymin=-2, ymax=7, % Adjust y-axis limits
    axis lines=middle,
		unbounded coords=jump,
    enlargelimits=true,
]

% Plot the integral curve using numerical integration
\addplot [blue, domain=0:15, samples=200] {trapz(0,x,10)};
\addplot [red, thick] {int((1)/(sqrt(1+x^2+ln(x)^2)))};
\end{axis}
\node[blue, anchor=west] at (2,5) {$\text{trapez}(a, b, n) = \frac{b-a}{2n} \left( f(a) + 2\sum_{i=1}^{n-1} f\left(a + \frac{i(b-a)}{n}\right) + f(b) \right)$};
\node[red, anchor=west] at (2,4) {$F(x) = \int_0^x \frac{1}{\sqrt{1+t^2+\ln^2 t}} dt$};
\node[red, anchor=west] at (2,3) {$F(0) = 1$};
\end{tikzpicture}
\end{document}