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

Abstract

The non-elementary integrals $\text{Si}_{β,α}=\int [\sin{(λx^β)}/(λx^α)] dx,β\ge1,α\leβ+1$ and $\text{Ci}_{β,α}=\int [\cos{(λx^β)}/(λx^α)] dx, β\ge1, α\le2β+1$, where $\{β,α\}\in\mathbb{R}$, 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 $\int [\sin^n{(λx^β)}/(λx^α)] dx$ and $\int [\cos^n{(λx^β)}/(λx^α)] dx$, where $n$ is a positive integer, are expressed in terms $\text{Si}_{β,α}$ and $\text{Ci}_{β,α}$, and then evaluated. $\text{Si}_{β,α}$ and $\text{Ci}_{β,α}$ 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 $\text{Ei}_{β,α}=\int (e^{λx^β}/x^α) dx$ where $β\ge1$ and $α\leβ+1$ and the logarithmic integral $\text{Li}=\int_μ^{x} dt/\ln{t}, μ>1$ are also expressed in terms of $_{2}F_2$, and their asymptotic expressions are investigated. It is found that for $x\ggμ$, $\text{Li}\sim {x}/{\ln{x}}+\ln{\left(\frac{\ln{x}}{\lnμ}\right)}-2-\lnμ\hspace{.075cm} _{2}F_{2}(1,1;2,2;\lnμ)$, where the term $\ln{\left(\frac{\ln{x}}{\lnμ}\right)}-2-\lnμ\hspace{.075cm} _{2}F_{2}(1,1;2,2;\lnμ)$ is added to the known expression in mathematical literature $\text{Li}\sim {x}/{\ln{x}}$.