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 \(_{1}F_1\) by expanding the integrand as a Taylor series. 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 \(_1F_2\), 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 \(\int e^{\lambda x^\alpha} dx, \alpha\ge 2\) such as the Gaussian distribution and the Maxwell-Bortsman distribution are given.

Keywords

• Non-elementary integral, Hypergeometric function, Confluent hypergeometric function, Asymptotic evaluation, Fundamental theorem of calculus, Gaussian, Maxwell-Bortsman distribution