EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART I
Abstract
The non-elementary integrals Siβ,α=∫[sin(λxβ)/(λxα)]dx, β≥1, α≤β+1 and Ciβ,α=∫[cos(λxβ)/(λxα)]dx, β≥1, α≤2β+1, where {β,α}∈R, are evaluated in terms of the hypergeometric functions 1F2 and 2F3, and their asymptotic expressions for |x|≫1 are also derived. The integrals of the form ∫[sinn(λxβ)/(λxα)]dx and ∫[cosn(λxβ)/(λxα)]dx, where n is a positive integer, are expressed in terms Siβ,α and Ciβ,α, and then evaluated. Siβ,α and Ciβ,α are also evaluated in terms of the hypergeometric function 2F2. And so, the hypergeometric functions, 1F2 and 2F3, are expressed in terms of 2F2. The exponential integral Eiβ,α=∫(eλxβ/xα)dx where β≥1 and α≤β+1 and the logarithmic integral Li=∫xμdt/lnt, μ>1, are also expressed in terms of 2F2, and their asymptotic expressions are investigated. For instance, it is found that for x≫2, Li∼x/lnx+ln(lnx/ln2)−2−ln22F2(1,1;2,2;ln2), where the term ln(lnx/ln2)−2−ln22F2(1,1;2,2;ln2) is added to the known expression in mathematical literature Li∼x/lnx. 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.
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