Int{x=0 to infinity}(exp(-x)*Ln(x)*Ln(x)dx)

[mathslover]

I have tried many hours on the following integrals and would appreciate any help from you.


1. Int{x=0 to infinity}(exp(-x)*Ln(x)*Ln(x)dx)

2. Int{x=0 to Infinity}(exp(-x*x)*Ln(x)dx)



Any idea guys?

 

[CompuChip]

Quickly checked it with Mathematica... the indefinite integrals are no fun (they involve hypergeometric functions and error functions) but the definite integrals come out relatively nicely.

I suppose you will want to somehow use
$$\int_0^\infty e^{-t} \ln(t) \, \mathrm dt = -\gamma$$
(the Euler gamma). The second one will also involve a Gaussian integral.

Will think a bit more...

 

[HallsofIvy]

This has nothing to do with "number theory". I am moving it to Calculus and Analysis.

 

[mathslover]

For the first integral ,

it can be shown to be = Gamma''(1) and

- Euler's Constant = Gamma'(1)

where Gamma(x)=Gamma Integral

I just don't know how to use the above facts.