ABSTRACT
This is the third in a series of papers dealing with the evaluation of definite integrals in the table of Gradshteyn and Ryzhik [40]. We consider here problems of the form ∫ ∞
e−tx P (lnx) dx, (3.1.1)
where t > 0 is a parameter and P is a polynomial. In future work we deal with the finite interval case ∫ b
e−tx P (lnx) dx, (3.1.2)
where a, b ∈ R+ with a < b and t ∈ R. The classical example∫ ∞ 0
e−x lnx dx = −γ, (3.1.3)
where γ is Euler’s constant is part of this family. The integrals of type (3.1.1) are linear combinations of
Jn(t) :=
e−tx (lnx)n dx. (3.1.4)
The values of these integrals are expressed in terms of the gamma function
Γ(s) =
xs−1e−x dx (3.1.5)
and its derivatives.
