What is $\int\frac {dx} {x!}$?

Question

if we assumed that $x$ is integer then it should be to integrat $\int\frac {dx} {x!}$ as $\int\frac {dx} {\Gamma(x+1)}$ and the latter is not known to me how to get it closed form, then I should ask in General :What is $\int\frac {dx} {x!}$ for $x$ as a real number ?

Answer 1

I doubt there's an explicit antiderivative for this. However a nice representation would be from Euler's reflection formula $\Gamma(1-z)\sin(\pi z)=\frac{\pi}{\Gamma(z)}$, which by analytic continuation should now hold for all $z$ (including integers). So:

$$\int_a^b\frac{1}{\Gamma(z+1)}dz=-\pi \int_a^b\Gamma(-z)\sin(\pi z)dz,$$

where you can now use your favorite integral representation of the gamma function (taking care about whether your bounds are negative or positive).