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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