EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART I - Victor Nijimbere (Ural Mathematical Journal, 2018)
The non-elementary integrals \(\text{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\)  \(\beta\ge1,\) \(\alpha\le\beta+1\) and \(\text{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\) \(\beta\ge1,\) \(\alpha\le2\beta+1\), 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 \(\text{Si}_{\beta,\alpha}\) 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 \(\beta\ge1\) 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. For instance, it is found that for \(x\gg2\), \(\text{Li}\sim {x}/{\ln{x}}+\ln{\left({\ln{x}}/{\ln{2}}\right)}-2- \ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})\), where the term \(\ln{\left({\ln{x}}/{\ln{2}}\right)}-2- \ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})\) is added to the known expression in mathematical literature \(\text{Li}\sim {x}/{\ln{x}}\). 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.