A type of multiple integral with log-gamma function

Abstract

In this paper, we give a general formula for the multiple integral

$$I={\int }_0^1{\int }_0^1\dots {\int }_0^{1}f\left(x_1+x_2+\cdots +x_n\right)d{x}_{1}d{x}_2\dots d{x}_n.$$

As an application, the integral $I$ with $f\left(x\right)=log\Gamma \left(x\right)$ is evaluated for all $n\in \mathbb{N}$. The subsidiary computational challenges are interesting in their own right.