What $\int_0^a x^ne^xdx$ looks like? Given the integral $$\int_0^ax^ne^xdx$$
I wanted to know what it looks like for an integer $n$ and a positive $a$. After evaluating some values with mathematica, I believe it looks like $e^ap_n(a)+c_n$, where $p_n(a)$ is a polynomial with degree $n$ in $a$, and $c_n$ is an integer. Is this correct?

\begin{align}
\int_0^a x^ne^xdx
=&\ \frac{d^n}{dt^n}\left(\int_0^ae^{t x}dx\right)_{t=1}
= \frac{d^n}{dt^n}\left(\frac{e^{ta}-1}t\right)_{t=1}\\
=& \ (-1)^n n! \bigg(e^a\sum_{k=0}^n \frac{(-a)^k}{k!} - 1\bigg)
\end{align}
