New Insights into Integrals Involving Gaussian Sums

In this article, we explore a series of elementary yet insightful results involving integrals related to Gaussian sums. Using techniques rooted in classical calculus, we derive several identities and evaluate nontrivial definite integrals that emerge naturally in this context. The approach is mostly elementary, avoiding the need for advanced machinery, and aims to shed light on the rich interplay between analysis and arithmetic structures arising from these integrals.