Definite Integral to Infinity of Exponential of -x by Logarithm of x

Theorem
Let $\ln t$ denote the natural logarithm function for real $t > 0$.

Let $e^{-t}$ denote the real exponential.

Then:


 * $\displaystyle \int_{0^+}^{ \mathop \to +\infty} \ln t \, e^{-t} \, \mathrm d t = - \gamma$

where the is an improper integral, and $\gamma$ is the Euler-Mascheroni Constant.