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

TL;DR

This paper derives explicit formulas for certain complex integrals involving sine, cosine, and exponential functions with power-law arguments, expressing them in terms of hypergeometric functions using series expansion methods.

Contribution

It provides new closed-form expressions for non-elementary integrals involving special functions, expanding the analytical tools available for such integrals.

Findings

Integrals are expressed in terms of hypergeometric functions $_{2}F_3$, $_{2}F_2$.

Series expansion method is used to evaluate the integrals.

Results extend the analytical solutions for complex non-elementary integrals.

Abstract

The non-elementary integrals $\mbox{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\beta\ge1,\alpha>\beta+1$ and $\mbox{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx, \beta\ge1, \alpha>2\beta+1$, where $\{\beta,\alpha\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric function $_{2}F_3$. On the other hand, the exponential integral $\mbox{Ei}_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx, \beta\ge1, \alpha>\beta+1$ is expressed in terms of $_{2}F_2$. The method used to evaluate these integrals consists of expanding the integrand as a Taylor series and integrating the series term by term.