Integral representation of the Euler-Mascheroni constant

[Millennial]

I am trying to prove a specific representation of Euler's constant, but I am not really getting anywhere. I hoped you could help me with this one, because I looked it up on the Internet and even though the relation itself is found in many webpages, its proof is in none. The relation is
$$\gamma=-\int_{0}^{\infty}e^{-t}\log(t)dt$$
I tried integrating by parts and integrating term by term using power series, but none of them show the identity. Thanks for your help from now.

 

[micromass]

See http://www.frm.utn.edu.ar/analisisdsys/material/funcion_gamma.pdf on page 15.

 

[Millennial]

Thanks for the article. It comes up with a pretty good proof using the Weierstrass product of the Gamma function.

I sincerely wonder how I could not find that one with two hours of searching.