Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (3.14)

\begin{equation*}
\int{\left( s{{x}^{s-1}}{\exp(-x)}-{{x}^{s}}{\exp(-x)} \right)}dx={{x}^{s}}{\exp(-x)}.
\end{equation*}Proof:
\begin{align*}
& \int{\left( s{{x}^{s-1}}\exp (-x)-{{x}^{s}}\exp (-x) \right)}dx \\
& =s{{x}^{s}}\exp (-x)h_{s-1}^{-x}-{{x}^{s+1}}\exp (-x)h_{s}^{-x} \\
& ={{x}^{s}}\exp (-x)\left( sh_{s-1}^{-x}-xh_{s}^{-x} \right) \\
& ={{x}^{s}}\exp (-x).
\end{align*}
$\square$

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