# Thread: Integral definition of the factorial function

1. ## Integral definition of the factorial function

Hello,
just out of mathematical curiosity, I found out on some website that the following statement holds for any $\displaystyle n > 0$ :

$\displaystyle \int_0^{\infty} \frac{x^n}{e^x} \ dx = n!$

How would one go to actually prove this statement ? Without going in too advanced mathematics if possible.

Thanks all

2. Originally Posted by Bacterius
Hello,
just out of mathematical curiosity, I found out on some website that the following statement holds for any $\displaystyle n > 0$ :

$\displaystyle \int_0^{\infty} \frac{x^n}{e^x} \ dx = n!$

How would one go to actually prove this statement ? Without going in too advanced mathematics if possible.

Thanks all
you opened a can of worms ...

The Gamma Function

3. Ah ... Well, I'll just keep this website in my favorites and come back to it in a couple of years
That is, when I will have learnt the basics of advanced calculus

4. You can integrate by parts n-times, or you can differentiate under the integral sign.

We know that $\displaystyle \int_{0}^{\infty} e^{-ax} \ dx = \frac{1}{a}$

And notice that $\displaystyle -\int^{\infty}_{0}\frac{\partial^{n}}{\partial a^{n}} \ e^{-ax} \ dx = \int_{0}^{\infty} x^{n}e^{-ax} \ dx$ (1)

Now switch the order of integration and differentiation (which in this case is allowed).

$\displaystyle -\frac{\partial^{n}}{\partial a^{n}}\int^{\infty}_{0}\ e^{-ax} \ dx = -\frac{\partial^{n}}{\partial a^{n}}\frac{1}{a} = n!a^{(-1-n)}$ (2)

Finally equate (1) and (2) and let a=1.