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

Abstract

The non-elementary integrals ${\text{Si}}_{\beta ,\alpha }=\int [\sin (\lambda x^{\beta })/(\lambda x^{\alpha })]dx,\beta \ge 1,\alpha \le \beta +1$ and ${\text{Ci}}_{\beta ,\alpha }=\int [\cos (\lambda x^{\beta })/(\lambda x^{\alpha })]dx,\beta \ge 1,\alpha \le 2\beta +1$, 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 $|x|\gg 1$ 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 ${\text{Si}}_{\beta ,\alpha }$ and ${\text{Ci}}_{\beta ,\alpha }$, and then evaluated. ${\text{Si}}_{\beta ,\alpha }$ and ${\text{Ci}}_{\beta ,\alpha }$ 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 ${\text{Ei}}_{\beta ,\alpha }=\int (e^{\lambda x^{\beta }}/x^{\alpha })dx$ where $\beta \ge 1$ and $\alpha \le \beta +1$ and the logarithmic integral $\text{Li}={\int }_{\mu }^{x}dt/\ln t,\mu >1$ are also expressed in terms of ${}_2{F}_2$, and their asymptotic expressions are investigated. It is found that for $x\gg \mu$, $\text{Li}\sim x/\ln x+\ln \left(\frac{\ln x}{\ln \mu }\right)-2-\ln \mu {}_2{F}_2(1,1;2,2;\ln \mu )$, where the term $\ln \left(\frac{\ln x}{\ln \mu }\right)-2-\ln \mu {}_2{F}_2(1,1;2,2;\ln \mu )$ is added to the known expression in mathematical literature $\text{Li}\sim x/\ln x$.

23 pages, 1 figure, Accepted for publication by the Ural Math. J

References in corpus (1)

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

Cited by in corpus (3)

• Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals: Part II
• Analytical valuation of some non-elementary integrals involving some exponential, hyperbolic and trigonometric elementary functions and derivation of new probability measures generalizing the gamma-type and normal distributions
• Evaluation of some non-elementary integrals involving the generalized hypergeometric function with some applications