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

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 $\text{Li}\sim {x}/{\ln{x}}$.