A Few Surprising Integrals

TL;DR

This paper demonstrates that specific families of integrals, which appear to depend on a parameter, are actually independent of it, using formulas related to stable vectors and the divide and color model.

Contribution

It introduces a novel approach to proving parameter independence of integrals using stable vector formulas and the divide and color model.

Findings

Certain integrals are independent of parameters.

Use of stable vector formulas simplifies integral analysis.

Application of the divide and color model aids in proofs.

Abstract

Using formulas for certain quantities involving stable vectors, due to I. Molchanov, and in some cases utilizing the so-called divide and color model, we prove that certain families of integrals which, ostensibly, depend on a parameter are in fact independent of this parameter.