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

Victor Nijimbere

Published 2017-02-25Version 1

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 $_1F_1$.