Atmospheric density derivation via canonical partition function

As the title points out i am trying to derive the atmospheric pressure. I understand the theory behind what i should do and how, but mathematically i have no idea how to proceed.

First of all, understanding: i have a particle in the atmosphere (we assume the atmosphere has a constant temperature). Now the different states of the air particle are the different heights of it from the surface. Because the temperature is constant, that means that the kinetic energy is a constant value while the potential energy changes with the height. Now in order to find the density at a height $h$ all i need to do is to find the probability of the particle being in that height.

So this is the probability of finding the particle in a height $h$ and $h + dh$:

$$dP(z)= 1/(2pihbar)^3 e^{-beta (vec p^2/2m + mgz)}dp_xdp_ydp_zdxdydz.$$

The first part in the above equation is the PDF of the continuous variable that is height.

Now when i try to integrate so that i can find the probability of the particle being in a height h i don’t know how to proceed:

$$P(z)= rho (z) = int 1/(2pihbar)^3 e^{-beta (vec p^2/2m + mgz)}dp_xdp_ydp_zdxdydz. $$

What should the the boundaries for $x$ and $y$? For $p_x$ $p_y$ and $p_z$ i can put from minus to plus infinity, but when i need to integrate regarding $x$ and $y$ i will get zero.
How do i solve this? In order to reach to this:

$$ rho (z) = rho(0)e^{-beta M g z}.$$