Evaluation of the non-elementary integral $\int e^{\lambda x^{\alpha }}dx,\alpha \ge 2$, and other related integrals

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 ${}_1{F}_1$. 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 ${}_1{F}_1$ and another one in terms of the hypergeometric function ${}_1{F}_2$, 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 $\int \sin (\lambda x^{\alpha })dx$, $\lambda \in \mathbb{C},\alpha \ge 2$. And the hypergeometric function ${}_1{F}_2$ is expressed in terms of the confluent hypergeometric function ${}_1{F}_1$. Some of the applications of the non-elementary integral $\int e^{\lambda x^{\alpha }}dx,\alpha \ge 2$ such as the Gaussian distribution and the Maxwell-Bortsman distribution are given.

15 pages, 1 figure

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