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

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.