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

The Gaussian integral, denoted as ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}dx$, 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 ${\int }_0^{\mathrm{\infty }}{e}^{-x^n}dx$, and two distinct types of Gaussian-like integrals: 1. Type I: ${\int }_0^{\mathrm{\infty }}{e}^{-f(x{)}^2}dx$, and 2. Type II: ${\int }_0^{\mathrm{\infty }}{e}^{-x^2}f(x)dx$, 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.