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 Ural Math. J., 2018, Volume 4, Issue 1, Pages 43–55 (Mi umj54)

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 ${Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,$ $\beta\ge1,$ $\alpha>\beta+1$ and ${Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,$ $\beta\ge1,$ $\alpha>2\beta+1$, where $\{\beta,\alpha\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric function $_{2}F_3$. On the other hand, the exponential integral ${Ei}_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx,$ $\beta\ge1,$ $\alpha>\beta+1$ 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.

DOI: https://doi.org/10.15826/umj.2018.1.004

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Citation: Victor Nijimbere, “Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals: part II”, Ural Math. J., 4:1 (2018), 43–55

Citation in format AMSBIB
\Bibitem{Nij18} \by Victor~Nijimbere \paper Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals: part II \jour Ural Math. J. \yr 2018 \vol 4 \issue 1 \pages 43--55 \mathnet{http://mi.mathnet.ru/umj54} \crossref{https://doi.org/10.15826/umj.2018.1.004} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=MR3848663} \elib{http://elibrary.ru/item.asp?id=35339281}