Evaluation of the non-elementary integral ${\int e^{\lambda x^\alpha} dx}$, ${\alpha\ge2}$ and other related integrals

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.