Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (4.2)

\begin{equation*}
{{E}_{(n)}}\left( x \right) ={{x}^{n-1}}\int_{x}^{\infty }{{{t}^{-n}}{\exp(-t)}dt}.
\end{equation*}Proof:
Using the transformation of variables, we derive
\begin{align*}
{{E}_{(n)}}\left(x\right) & =\int_{1}^{\infty }{\frac{\exp \left( -xt \right)}{{{t}^{n}}}}dt \\& =\int_{x}^{\infty }{{{x}^{n-1}}\frac{\exp \left( -u \right)}{{{u}^{n}}}}du \\
& ={{x}^{n-1}}\int_{x}^{\infty }{{{t}^{-n}}\exp \left( -t \right)}dt.
\end{align*}
$\square$

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