Evaluation of the non-elementary integral $\int e^{λx^α} dx, α\ge2$, and other related integrals

Abstract

A formula for the non-elementary integral $\int e^{λx^α} dx$ where $α$ 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 $α= 2$, 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, $\int \cosh(λx^α)dx$, $\int \sinh(λx^α)dx$, $\int \cos(λx^α)dx$ and $\int \sin(λx^α)dx$, $λ\in \mathbb{C}, α\ge2$. 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 $\int e^{λx^α}dx,α\ge2$ such as the Gaussian distribution and the Maxwell-Bortsman distribution are given.