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

Abstract: The Gaussian integral, denoted as ( \int_{-\infty}^{\infty} e^{-x^2} dx ), 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 ( \int_{0}^{\infty} e^{-x^n} dx ), and two distinct types of Gaussian-like integrals: 1. Type I: ( \int_{0}^{\infty} e^{-f(x)^2} dx ), and 2. Type II: ( \int_{0}^{\infty} e^{-x^2} f(x) dx ), where f(x) is a continuous function. The study of integrals related to Gaussian-like functions has been explored in the work of Huang and Dominy \cite{Dnd}. Our approach to evaluating these integrals relies on specialized functions, including error functions, complementary error functions, imaginary error functions, and Basel functions.