I'm trying to solve an integral with Mathematica, but I have no idea, what I have to input. The integral is as followed: $$ \int d^{3N}\!p~e^{-\frac{\beta}{2m}\vec p^2} $$ I've already tried this

integrate[e^(-a*{p1,p2,p3}^2),{{p1,p2,p3}^N,-Infinity,Infinity}]

but its not working at all. I guess, I have to determine the vector before, or?

I'm trying to compute a multidimensional integral with a variable number of dimensions.

The integral is as follows: $$ \int d^{3N}\!p~e^{-\frac{\beta}{2m}\vec p^2}. $$

I have tried this

Integrate[e^(-a*{p1,p2,p3}^2),{{p1,p2,p3}^N,-Infinity,Infinity}]

but it's not working at all. I guess I have to determine the vector before, or?

More generally, to what extent can Mathematica compute such integrals for arbitrary $N$ and arbitrary integrands in place of $e^{-\frac{\beta}{2m}\vec p^2}$?