# Evaluation of the non-elementary integral $\int e^{\lambda x^\alpha} dx, \alpha\ge2$, and other related integrals

## TL;DR

This paper derives formulas for non-elementary integrals involving exponential functions with power-law exponents using hypergeometric functions, verifying results through classical Gaussian integrals and exploring applications in probability distributions.

## Contribution

It provides new explicit expressions for these integrals in terms of confluent hypergeometric functions, expanding the analytical tools for related mathematical and physical problems.

## Key findings

- Derived formulas for integrals in terms of $_1F_1$ and $_1F_2$
- Verified formulas using Gaussian distribution properties
- Connected hypergeometric functions to applications in distributions

## Abstract

A formula for the non-elementary integral $\int e^{\lambda x^\alpha} dx$ where $\alpha$ is real and greater or equal two, is obtained in terms of the confluent hypergeometric function $_1F_1$. This result is verified by directly evaluating the area under the Gaussian Bell curve, corresponding to $\alpha = 2$, using the asymptotic expression for the confluent hypergeometric function and the Fundamental Theorem of Calculus (FTC). Two different but equivalent expressions, one in terms of the confluent hypergeometric function $_1F_1$ and another one in terms of the hypergeometric function $_1F_2$, are obtained for each of these integrals, $\int \cosh(\lambda x^\alpha)dx$, $\int \sinh(\lambda x^\alpha)dx$, $\int \cos(\lambda x^\alpha)dx$ and $\int \sin(\lambda x^\alpha)dx$, $\lambda\in \mathbb{C}, \alpha\ge2$. And the hypergeometric function $_1F_2$ is expressed in terms of the confluent hypergeometric function $_1F_1$. Some of the applications of the non-elementary integral $\int e^{\lambda x^\alpha}dx,\alpha\ge2$ such as the Gaussian distribution and the Maxwell-Bortsman distribution are given.