A half-automated study of a 2-parameter family of integrals

TL;DR

This paper presents a semi-automated approach combining analytical methods, CAS computations, and integer sequence analysis to study a family of parametric integrals, revealing connections to Catalan numbers.

Contribution

It introduces a general algorithm for computing a class of parametric integrals and explores their relationship with special functions and classical number sequences.

Findings

Connected the integral to Catalan numbers

Developed a general computational algorithm

Used CAS for specific parameter evaluations

Abstract

The study of some parametric integrals is presented with a combined approach of analytical development, the usage of a Computed Algebra System (CAS) and of the Online Encyclopedia of Integer Sequences. The methodology for the solution includes a) an analytical investigation for the study of the parametric integral, b) computations with a CAS of the integral for specific values of the parameter, c) investigation of the connection between the integral and special functions or classical numbers, and d) derivation of a general algorithm for the complete computation of the parametric integral. The central example of the paper is the parametric integral \begin{equation*} \label{the general integral} I_{n}^{(p)}=\int_0^{\pi /4} x^p \tan^n x\; dx, \end{equation*} The work reveals a connection of this parametric integral with Catalan numbers.