Can I Write It As An Integral?

Question

Is there a way to write $$\Pi _{i=1}^n (i)$$ as an integral?

[ $\because \Pi (x) = \Gamma (x+1) $ and $\Gamma (x+1) $can be written in form of an integral , that's why i am asking, can we write above equation also? If yes, how?]

Answer 1

We have that for $n \in \mathbb{N}$, $$P_n=\prod_{i=1}^ni=n!=\Gamma(n+1)$$ and since for $x > 0$ $$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt$$ we also can say that $$\prod_{i=1}^n i=\int_0^\infty t^ne^{-t}dt$$