Solution
= Solution
The classical one-particle phase-space <integral> is
$$
Z_1=\frac1{h^3}\int_{\mathbb R^3}\!d^3x
\int_{\mathbb R^3}\!d^3p\,
e^{-\beta(pc+U(x))}.
$$
The <momentum> <integral> is
$$
4\pi\int_0^\infty p^2e^{-\beta cp}\,dp
=\frac{8\pi}{\beta^3c^3}.
$$
Hence
$$
\boxed{
Z_1=\frac{8\pi(k_BT)^3}{h^3c^3}
\int_{\mathbb R^3}e^{-U(x)/(k_BT)}\,d^3x
=\frac{(k_BT)^3}{\pi^2(\hbar c)^3}
\int e^{-U/(k_BT)}\,d^3x.}
$$