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

Abstract

The non-elementary integrals $\mbox{Si}_{β,α}=\int [\sin{(λx^β)}/(λx^α)] dx,β\ge1,α>β+1$ and $\mbox{Ci}_{β,α}=\int [\cos{(λx^β)}/(λx^α)] dx, β\ge1, α>2β+1$, where $\{β,α\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric function $_{2}F_3$. On the other hand, the exponential integral $\mbox{Ei}_{β,α}=\int (e^{λx^β}/x^α) dx, β\ge1, α>β+1$ 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.