TITLE: Integral of factorial function
QUESTION [6 upvotes]: What can we say about the integral $\displaystyle\int_{0}^{a} x! dx$?
Or something like $\displaystyle\int_{0}^{3} x! dx$?

REPLY [6 votes]: This is a very amazing problem. As I said earlier, I did not find any closed form for the integral and then I only performed numerical integrations.
What is surprizing if that $y(a)=\displaystyle\int_{0}^{a} \Gamma(1+x)~ dx$ looks very much to $x(a)=\Gamma(1+a)$ (plot the two curves as a function of $a$).
Taking into account Lucian's comment, I performed a parametric plot  and I observed that, for values of $a \gt 1$, $y$ is almost linear with $x$.
A regression $\log(y)=a +b \log(x)$, for the range $2 \leq a \leq 12$, leads to $a=-0.0775761$ and $b=0.949784$ with $R^2=0.999467$.