An Analysis of the Generalized Gaussian Integrals and Gaussian Like Integrals of Types I and II

The Gaussian integral, denoted as (mathematical expression), plays a significant role in mathematical literature. In this paper, we explore a family of integrals related to Gaussian functions. Specifically, we introduce generalized Gaussian integrals, represented as (mathematical expression), and two distinct types of Gaussian-like integrals: 1. Type I: (mathematical expression), and 2. Type II: (mathematical expression), where f(x) is a continuous function. The study of integrals related to Gaussian-like functions has been explored in the work of Huang [8] and Dominy [7]. Our approach to evaluating these integrals relies on specialized functions, including error functions, complementary error functions, imaginary error functions, and Basel functions.