Is it possible to evaluate an integral using the xfp package?

Question

I am curious if it is possible to evaluate an integral using the xfp package using tex language?

\fpeval{\int_{-1}^{1}\frac{1}{\sqrt{1 - x^2}}dx}}

Just throwing out the idea. I know I can easily do this by python/Matlab/mathematica/calculator/hand but I love LaTeX.

Answer 1

Modulo coding errors this is a 6 step simpsons rule numeric approximation (not coded very efficiently as the end points of each step are evaluated twice)

l3fp could get a more accurate result than this if coded more carefully, but this is enough for this morning:-)

\documentclass{article}
\usepackage{xfp}
\def\fx#1{(1/(sqrt(1 - (#1)^2)))}
\def\simp#1#2#3{((((#2)-(#1))/6)*(#3{#1}+ 4*#3{(#1+(#2))/2} + #3{#2}))}
\begin{document}
\typeout{\fpeval{
2*(
 \simp{-0.99}{-0.9}{\fx}
+\simp{-0.9}{-0.8}{\fx}
+\simp{-0.8}{-0.6}{\fx}
+\simp{-0.6}{-0.4}{\fx}
+\simp{-0.4}{-0.2}{\fx}
+\simp{-0.2}{0}{\fx}
)
}}
\end{document}

producing

2.888405590181524

Which isn't quite the same as this:

Answer 2

Here is a solution based on the pst-ode package. The integrand is symmetrical about x=0, so integration can be started at 0 and the result is multiplied by 2.

Typeset with pdflatex --shell-escape:

\documentclass{article}
\pagestyle{empty}

%%%%%%%%%%%%%%%%%%%%%%% solve ODE in auxiliary document %%%%%%%%%%%%%%%%%%%%%%%%
\begin{filecontents}[overwrite]{solve.tex}
\documentclass{article}
\usepackage{pst-ode}

\begin{document} 
% arguments:
%   algebraicAll --> all arguments in algebraic notation
%   saveData     --> also write result into file `result.dat'
%   `result'     --> PostScript variable that takes result
%   2*y[0]       --> output format in `result' and `result.dat'
%   0, 1         --> integration interval t_0, t_e
%   2            --> number of saved output points t_0, t_e
%   0            --> initial value
%   1/sqrt(...)  --> right-hand side of ODE 
\pstODEsolve[algebraicAll,saveData]{result}{2*y[0]}{0}{0.9999999}{2}{0}{ 
  1 / sqrt(1-t^2)
}
dummy text
\end{document}
\end{filecontents}

\immediate\write18{latex solve}
\immediate\write18{dvips solve}
\immediate\write18{ps2pdf -dNOSAFER solve.ps}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newread\resfile\immediate\openin\resfile=result.dat
\immediate\read\resfile to \dummy  % read and throw away initial value
\immediate\read\resfile to \result % read definite integral value

\begin{document}

\[
  \int_{-1}^{1} \frac{1}{\sqrt{1 - x^2}} dx \approx \result
\]

\end{document}