Some integrals related to the Basel problem

TL;DR

This paper evaluates parameter-dependent arctangent and logarithmic integrals, deriving closed-form series and integral representations for classical constants, thus advancing the understanding of related mathematical series and constants.

Contribution

It introduces new evaluations of integrals involving arctangent and logarithms that lead to novel series and constant representations.

Findings

Closed-form summations of specific series

Integral representations of classical constants

New relationships between integrals and series

Abstract

We evaluate several arctangent and logarithmic integrals depending on a parameter. This provides a closed form summation of certain series and also gives integral and series representation of some classical constants.