Explicit evaluation of some integrals involving polylogarithm functions

TL;DR

This paper provides explicit evaluations of certain integrals involving polylogarithm functions and logarithms, offering new formulas and extending previous results in special function analysis.

Contribution

The paper introduces explicit formulas for integrals involving polylogarithms and logarithms, expanding the analytical tools available for these special functions.

Findings

Explicit formulas for integrals of the form ∫₀ˣ t^m Li_p(t) dt

Explicit formulas for integrals of the form ∫₀ˣ log^m(t) Li_p(t) dt

Additional derived integrals involving logarithmic functions

Abstract

In this paper, we give explicit evaluation for some integrals involving polylogarithm functions of types $\int_{0}^{x}t^{m} Li_{p}(t)\mathrm{d}t$ and $\int_{0}^{x}\log^{m}(t) Li_{p}(t)\mathrm{d}t$. Some more integrals involving the logarithm function will also be derived.