Integrate Factorial Straight to question, how do you analytically integrate factorial function like the following: $\int_0^\infty 1/(n-1)!\, \mathrm dn\ $. This is equivalent to $\int_0^\infty 1/Γ(n)\, \mathrm dn\ $ but how do you integrate something like this. Using grapher shows a nice smooth graph that converges quickly.

**A hint.**

Your integral is:

$$I=\int_0^\infty \frac{dx}{\Gamma(x)}$$

By reflection formula for the Gamma function:

$$\frac{1}{\Gamma(x)}=\frac{1}{\pi} \Gamma(1-x) \sin \pi x$$

$$I=\frac{1}{\pi} \int_0^\infty \Gamma(1-x) \sin \pi x~ dx$$

Now by the integral definition of the Gamma function we get:

$$\Gamma(1-x)= \int_0^\infty \frac{e^{-t}}{t^x}dt$$

So now the integral becomes:

$$I=\frac{1}{\pi} \int_0^\infty \int_0^\infty \sin (\pi x)~ \frac{e^{-t}}{t^x}~dt ~ dx$$

$$I=\frac{1}{\pi} \int_0^\infty \int_0^\infty \sin (\pi x)~ e^{- \ln(t) x} e^{-t}~dt ~ dx$$

See @nospoon's comment. No closed form is available, apparently, but maybe this hint might still help.