How would I evaluate $\int{x!}dx$? Obviously elementary functions will not suffice. I looked up this problem on Google and have found no results. I've tried to logically think about this and haven't reached any conclusions. Obviously, the antiderivative of $x!$ implies that there must be a function whose derivative is $x!$. I've looked at the graph of $x!$ and it seems like you can cut the factorial at $x = 0$. The positive side resembles an exponential graph, while the negative side is more complicated. Is there any way to approach this?
(Also, this isn't for an assignment or anything, I was just interested in whether there is a way to take the antiderivative of a factorial. I'm also only in AP Calculus BC so I'm not an expert in this.)
If you want to consider $x!$ as a step function, this integrals boils down to
\begin{gather} \sum_{i=1}^n i! \end{gather}
which can be rewritten in terms of recurrence relations as $f(n) = f(n-1) + n!$ for which the solution is $f(n) = (-1)^{n + 1} Γ(n + 2)\ \ !(-n - 2)\, +\, !(-2)$ where $!x$ is the subfactorial function and $Γ$ is the gamma function (the extension of the factorial to the reals)