Evaluation of some non-elementary integrals of sine, cosine and exponential integrals type

摘要

The non-elementary integrals $\text{Si}_{\beta,\alpha}=\int [\sin{(\lambdax^\beta)}/(\lambda x^\alpha)] dx$ and $\text{Ci}_{\beta,\alpha}=\int[\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx$, where $\beta\ge1$ and$\alpha\ge1$, are evaluated in terms of the hypergeometric functions $_{1}F_2$and $_{2}F_3$ respectively, and their asymptotic expressions for $|x|\gg1$ arederived. Integrals of the form $\int [\sin^n{(\lambda x^\beta)}/(\lambdax^\alpha)] dx$ 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. On the other hand, $\text{Si}_{\beta,\alpha}$ and $\text{Ci}_{\beta,\alpha}$are evaluated in terms of the hypergeometric function $_{2}F_2$. And so, thehypergeometric functions, $_{1}F_2$ and $_{2}F_3$, are expressed in terms of$_{2}F_2$. The integral $\text{Ei}_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha)dx$ where $\beta\ge1$ and $\alpha\ge1$, and the logarithmic integral$\text{Li}=\int_{2}^{x} dt/\ln{t}$, are expressed in terms of $_{2}F_2$, andtheir asymptotic expressions are investigated. It is found that $\text{Li}\sim{x}/{\ln{x}}+\ln{\left(\frac{\ln{x}}{\ln{2}}\right)}-2- \ln{2}\hspace{.075cm}_{2}F_{2}(1,1;2,2;\ln{2})$, where the term$\ln{\left(\frac{\ln{x}}{\ln{2}}\right)}-2- \ln{2}\hspace{.075cm}_{2}F_{2}(1,1;2,2;\ln{2})$ is negligible if $x\sim O(10^6)$ or higher.