Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals: Part I (1703.01907v2)

Abstract: The non-elementary integrals $\text{Si}{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\beta\ge1,\alpha\le\beta+1$ and $\text{Ci}{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx, \beta\ge1, \alpha\le2\beta+1$, where ${\beta,\alpha}\in\mathbb{R}$, are evaluated in terms of the hypergeometric functions ${1}F_2$ and ${2}F_3$, and their asymptotic expressions for $|x|\gg1$ are also derived. The integrals of the form $\int [\sin^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx$ and $\int [\cos^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx$, where $n$ is a positive integer, are expressed in terms $\text{Si}{\beta,\alpha}$ and $\text{Ci}{\beta,\alpha}$, and then evaluated. $\text{Si}{\beta,\alpha}$ and $\text{Ci}{\beta,\alpha}$ are also evaluated in terms of the hypergeometric function ${2}F_2$. And so, the hypergeometric functions, ${1}F_2$ and ${2}F_3$, are expressed in terms of ${2}F_2$.The exponential integral $\text{Ei}{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx$ where $\beta\ge1$ and $\alpha\le\beta+1$ and the logarithmic integral $\text{Li}=\int{\mu}^{x} dt/\ln{t}, \mu>1$ are also expressed in terms of ${2}F_2$, and their asymptotic expressions are investigated. It is found that for $x\gg\mu$, $\text{Li}\sim {x}/{\ln{x}}+\ln{\left(\frac{\ln{x}}{\ln{\mu}}\right)}-2-\ln{\mu}\hspace{.075cm} _{2}F{2}(1,1;2,2;\ln{\mu})$, where the term $\ln{\left(\frac{\ln{x}}{\ln{\mu}}\right)}-2-\ln{\mu}\hspace{.075cm} {2}F{2}(1,1;2,2;\ln{\mu})$ is added to the known expression in mathematical literature $\text{Li}\sim {x}/{\ln{x}}$.