EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART II

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.