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 $\text{ Li}\sim { x}/{ \ln{ x}}$.
Categories: math.CA