
Evaluation of some nonelementary integrals involving sine, cosine, exponential and logarithmic integrals: part I
Victor Nijimbere^{} ^{} School of Mathematics and Statistics, Carleton University,
Ottawa, Ontario, Canada
Abstract:
The nonelementary integrals $Si_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,$ $\beta\ge1,$ $\alpha\le\beta+1$ and $Ci_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,$ $\beta\ge1,$ $\alpha\le2\beta+1$, where $\{\beta,\alpha\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric functions $_{1}F_2$ and $_{2}F_3$, and their asymptotic expressions for $x\gg1$ are also derived. The integrals of the form $\int [\sin^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx$ and $\int [\cos^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx$, where $n$ is a positive integer, are expressed in terms $Si_{\beta,\alpha}$ and $Ci_{\beta,\alpha}$, and then evaluated. $Si_{\beta,\alpha}$ and $Ci_{\beta,\alpha}$ are also evaluated in terms of the hypergeometric function $_{2}F_2$. And so, the hypergeometric functions, $_{1}F_2$ and $_{2}F_3$, are expressed in terms of $_{2}F_2$.
The exponential integral $Ei_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx$ where $\beta\ge1$ and $\alpha\le\beta+1$ and the
logarithmic integral $Li=\int_{\mu}^{x} dt/\ln{t}$, $\mu>1$, are also expressed in terms of $_{2}F_2$, and their asymptotic expressions are investigated. For instance, it is found that for $x\gg2$, $Li\sim {x}/{\ln{x}}+\ln{({\ln{x}}/{\ln{2}})}2
\ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})$, where the term $\ln{({\ln{x}}/{\ln{2}})}2
\ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})$ is added to the known expression in mathematical literature $Li\sim {x}/{\ln{x}}$.
The method used in this paper consists of expanding the integrand as a Taylor and integrating the series term by term, and can be used to evaluate the other cases which are not considered here. This work is motivated by the applications of sine, cosine exponential and logarithmic integrals in Science and Engineering, and some applications are given.
Keywords:
Nonelementary integrals, Sine integral, Cosine integral, Exponential integral, Logarithmic integral, Hyperbolic sine integral, Hyperbolic cosine integral, Hypergeometric functions, Asymptotic evaluation, Fundamental theorem of calculus.
DOI:
https://doi.org/10.15826/umj.2018.1.003
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Victor Nijimbere, “Evaluation of some nonelementary integrals involving sine, cosine, exponential and logarithmic integrals: part I”, Ural Math. J., 4:1 (2018), 24–42
Citation in format AMSBIB
\Bibitem{Nij18}
\by Victor~Nijimbere
\paper Evaluation of some nonelementary integrals involving sine, cosine, exponential and logarithmic integrals: part I
\jour Ural Math. J.
\yr 2018
\vol 4
\issue 1
\pages 2442
\mathnet{http://mi.mathnet.ru/umj53}
\crossref{https://doi.org/10.15826/umj.2018.1.003}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=MR3848662}
\elib{https://elibrary.ru/item.asp?id=35339280}
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Cycle of papers
 Evaluation of some nonelementary integrals involving sine, cosine, exponential and logarithmic integrals: part I
Victor Nijimbere Ural Math. J., 2018, 4:1, 24–42
 Evaluation of some nonelementary integrals involving sine, cosine, exponential and logarithmic integrals: part II
Victor Nijimbere Ural Math. J., 2018, 4:1, 43–55

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