Integral of factorial function
What can we say about the integral $\displaystyle\int_{0}^{a} x! dx$?
Or something like $\displaystyle\int_{0}^{3} x! dx$?
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$\begingroup$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 $R^2=0.999467$.
$\endgroup$ 2 $\begingroup$The factorial function is only defined on the positive integers, so those don't make sense. However, there is a generalization of the factorial called the Gamma function which you might want to check out.
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