I derived the expression

\begin{equation}

\int\limits_{0}^{\infty} \mathrm{e}^{-x} \mathrm{ln}(x) \mathrm{d} x = -\gamma

\label{eq:1608161}

\tag{1}

\end{equation}

for the Euler-Mascheroni constant here. However, there is a far easier method that was fully derived in *Advanced Integration Techniques* by Zaid Alyafeai. I recommend this book to readers of this blog. It is free and contains many useful and interesting results.

We start with

\begin{equation}

\int\limits_{0}^{\infty} \mathrm{e}^{-x} x^t \mathrm{d} x = \Gamma(t+1)

\label{eq:1608162}

\tag{2}

\end{equation}

Differentiate with respect to \(t\)

\begin{equation}

\int\limits_{0}^{\infty} \mathrm{e}^{-x} x^t \mathrm{ln}(x) \mathrm{d} x = \frac{d\Gamma(t+1)}{dt} = \Gamma(t+1) \psi^{(0)}(t+1)

\label{eq:1608163}

\tag{3}

\end{equation}

Taking the limit of equation \eqref{eq:1608163}, \(t \to 0\) yields

\begin{equation}

\int\limits_{0}^{\infty} \mathrm{e}^{-x} \mathrm{ln}(x) \mathrm{d} x = \Gamma(1) \psi^{(0)}(1) = -\gamma

\label{eq:1608164}

\tag{4}

\end{equation}

Hey thanks for recommending myou book.

I was happy to do so and I look forward to the new addition that you mentioned on your website.

Yeah I added around 50 more pages. It should be out after maybe two weeks.

When it is ready, let me know at appliedclassicalanalysis AT gmail DOT com and I will create a blog post with a link.

https://zaidalyafeai.files.wordpress.com/2015/09/advanced-integration-techniques6.pdf