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Evaluation of the non-elementary integral ${\int e^{\lambda x^\alpha} dx}$, ${\alpha\ge2}$ and other related integrals
Ural mathematical journal, Tome 3 (2017) no. 2, pp. 130-142 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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Victor Nijimbere. Evaluation of the non-elementary integral ${\int e^{\lambda x^\alpha} dx}$, ${\alpha\ge2}$ and other related integrals. Ural mathematical journal, Tome 3 (2017) no. 2, pp. 130-142. http://geodesic.mathdoc.fr/item/UMJ_2017_3_2_a13/

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