Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (4.1)

\begin{equation*}
{{E}_{(1)}}\left( x \right) =\int_{x}^{\infty }{{{t}^{-1}}{\exp(-t)}dt}.
\end{equation*}Proof:
Let $u=xt$, $du=xdt$, and $t=u/x$. We transform ${{E}_{(1)}}$ into an integral function of $u$
\begin{align*}
{{E}_{(1)}}\left( x \right) &=\int_{1}^{\infty }{\frac{\exp (-xt)}{t}dt} \\
& =\int_{x}^{\infty }{{{u}^{-1}}\exp (-u)du}.
\end{align*}
We can change the variable $u$ back to $t$ and conclude the proof.
$\square$

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