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

Victor Nijimbere     (School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada)

#### 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.

#### Keywords

Non-elementary integrals; Sine integral; Cosine integral; Exponential integral; Logarithmic integral; Hyperbolic sine integral; Hyperbolic cosine integral; Hypergeometric functions

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