# What is $\int x!$ $dx$?

What is $\int x!$ $dx$. $f(x)=x!$ looks something like this. Do we have any formula for finding this indefinite integral.

• Short answer: No. There is no formula. It does not make sense to take the integral of a discrete valued function. The Gamma Function on the other hand is a function that acts like the factorial function for all non-negative integers and also has values for a real $x$. So if you consider extending your factorial function $x!=\Gamma(x+1)$ then the indefinite integral of the Gamma function is something that can be discussed. Aug 28 '14 at 17:39

If you use the gamma function definition of factorial $$x! = \int_0^\infty y^{x}e^{-y} dy$$ and change the order of integration in the resulting double integral

$$\int_b^a x!dx = \int_b^a\int_0^\infty y^{x}e^{-y} dy dx = \int_0^\infty \int_b^a y^{x}e^{-y} dx dy$$

that might lead to a solution.

• $x! = \int_0^\infty y^x e^{-y} dy = \Gamma(y+1)$, you're off by one. Aug 28 '14 at 9:39
• Can you do it without using gamma function !! Aug 28 '14 at 9:41
• @NajibIdrissi thanks. Aug 28 '14 at 9:47
• @Abstraction Then it will make no sense! Aug 28 '14 at 9:48
• You are wrong. $x! \neq \Gamma(x)$. $x! = \Gamma(x+1)$, so the first integral is wrong because of it. Furthermore the domains are really not the same. Aug 28 '14 at 9:50

The question is weird, because the factorial function is defined for nonnegative integers by $n!=\prod_{k=1}^n k$ definition. That is why you rather could make a summation of it than an integral. The integration is not a tool for discrete functions.

On the other hand as @L'universo said there is a generalization of factorial function called Gamma function which is defined on the whole complex plane, so maybe you can use $\int \Gamma(t+1) dt$ this. But I don't know what to do with it.

• OP WolframAlpha plots the Gamma function in your input. You want to find the integral for factorial which wouldn't work because integrals are not defined for integers. Please consider what exactly you need. Aug 28 '14 at 9:50