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An integral with exponential tail

Posted: Mon Sep 19, 2016 10:51 am
by Riemann
Let $n \in \mathbb{N}$ and let us define the function $\displaystyle \mathcal{E}_{\tau, n} (x) = \int_{0}^{x} e^{-t} t^n \, {\rm d}t$ better known as lower incomplete gamma function. Prove that:

$$\bigintsss_{0}^{1}\left ( e^x - 1 - x - \frac{x^2}{2} - \cdots - \frac{x^n}{n} \right ) \, {\rm d}x = \frac{e}{n!} \mathcal{E}_{\tau, n} (1) - \frac{1}{n!(n+1)}$$