EVALUATION OF THE NON-ELEMENTARY INTEGRAL \(\int e^{λ x^α}dx\), \(α\ge 2\), AND OTHER RELATED INTEGRALS - Victor Nijimbere (Ural Mathematical Journal, 2017)
A formula for the non-elementary integral \(\int e^{λ x^α} dx\) where \(α\) 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 \(α=2\), 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 \(_{1}F_1\) and another one in terms of the hypergeometric function \(_1F_2\), are obtained for each of these integrals, \(\int\cosh(λ x^α)dx\), \(\int\sinh(λ x^α)dx\), \(\int \cos(λ x^α)dx\) and \(\int\sin(λ x^α)dx\), \(λ\in \mathbb{C},α\ge2\). 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 \(\int e^{λ x^α} dx, α\ge 2\) such as the Gaussian distribution and the Maxwell-Bortsman distribution are given.