Analytical valuation of some non-elementary integrals involving some exponential, hyperbolic and trigonometric elementary functions and derivation of new probability measures generalizing the gamma-type and normal distributions

Abstract

The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, $\int x^{\alpha }{e}^{\eta x^{\beta }}dx,\int x^{\alpha }\cosh \left(\eta x^{\beta }\right)dx,\int x^{\alpha }\sinh \left(\eta x^{\beta }\right)dx,\int x^{\alpha }\cos \left(\eta x^{\beta }\right)dx$ and $\int x^{\alpha }\sin \left(\eta x^{\beta }\right)dx$ where $\alpha ,\eta$ and $\beta$ are real or complex constants are evaluated in terms of the confluent hypergeometric function ${}_1{F}_1$ and the hypergeometric function ${}_1{F}_2$. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions ${}_1{F}_1$ and ${}_1{F}_2$. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and normal distributions are also obtained. The obtained generalized distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square (${\chi }^2$) statistical tests and those based on central limit theorem (CLT)).

23 pages, improved version

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