Convergence criteria for Frullani-type integrals involving differences of cosines

Abstract

For $p,q\in \mathbb{N}$ and $\hat{I}\pm ,{\hat{I}}^2\in \mathbb{R}$, we investigate the family of improper integrals \[\int_0^\infty\frac{(\cosαx-\cosβx)^p}{x^q}dx.\] We establish a complete classification of the parameter ranges $(p,q;\hat{I}\pm ,{\hat{I}}^2)$ for which the integrals converge or diverge, and we derive explicit closed-form evaluations in all convergent cases. The analysis also reveals a family of combinatorial identities arising naturally from coefficients in the trigonometric power expansions. As a further application of the same method, we study an analogous class of integrals involving powers of sine differences. This extends the work of Laoharenoo and Boonklurb in 2022.