How to write half integrals in LaTeX?

Question

I would like to know how to write half integrals and generally integrals of nth order in LaTeX, for example in Cauchy's formula for repeated integration, because I didn't find how to do that on the whole internet. Also I attached how I write this on paper.

Answer 1

Maybe you're looking for some specific notation that I don't know about, but in LaTeX, Cauchy's formula for repeated integration can for example be written as:

I^nf(x) = \int_0^x \int_0^{x_1} \cdots \int_0^{x_{n-1}} f(x_{n}) \, \mathrm{d}x_{n} \cdots \, \mathrm{d}x_2 \, \mathrm{d}x_1 = \frac{1}{(n-1)!} \int_0^x\left(x-t\right)^{n-1} f(t)\,\mathrm{d}t

Answer 2

Add a phantom integral, so the subscript is at the right place.

\documentclass{article}
\usepackage{amsmath}
\NewDocumentCommand{\INT}{m}{\mathop{{\vphantom{\int}}}\nolimits_{#1}!\int}
\begin{document}
A displayed integral
[
\INT{n}_a^b f(x),dx
]
A text style integral $\INT{n}_a^b f(x),dx$
\end{document}

Answer 3

It is possible to use also tensor package with \mathclap.

\documentclass{article}
\usepackage{tensor}
\usepackage{mathtools}
\begin{document}
[
\tensor[_{\mathclap{c}}]{\int}{_a^b} f(x) \ dx
]
$\tensor[_{\mathclap{c}}]{\int}{_a^b} f(x) \ dx$
\end{document}