$Z=\displaystyle\frac{1}{N!}\int\displaystyle\frac{\prod_i^Nd^3p_i\prod_i^Nd^3q_i}{h^{3N}}e^{-\beta(K+U)}$

Z=\displaystyle\frac{1}{N!}\int\displaystyle\frac{\prod_i^Nd^3p_i\prod_i^Nd^3q_i}{h^{3N}}e^{-\beta(K+U)}

$Z_K=\displaystyle\frac{1}{N!h^{3N}}\displaystyle\int\prod_i^Nd^3p_ie^{-\beta K}$

Z_K=\displaystyle\frac{1}{N!h^{3N}}\displaystyle\int\prod_i^Nd^3p_ie^{-\beta K}

$ Z_K =\displaystyle\frac{1}{ N !}\left(\displaystyle\frac{2 \pi m }{ h ^2 \beta }\right)^{3 N /2}$

Z_K =( 2* pi * m /( h ^2* beta )^(3* N /2)/factor( N )

$Z_U=\displaystyle\int\prod_i^Nd^3q_ie^{-\beta U}$

Z_U=\displaystyle\int\prod_i^Nd^3q_ie^{-\beta U}

$I(\beta)\equiv\displaystyle\int_0^{\infty}(e^{-\beta u(r)}-1)4\pi r^2dr$

I(\beta)\equiv\displaystyle\int_0^{\infty}(e^{-\beta u(r)}-1)4\pi r^2dr

$ \ln Z_U = N \ln V +\displaystyle\frac{1}{2}\displaystyle\frac{ N ^2}{ V } I( \beta )$

lnZ_U = N *ln( V )+ N ^2*I( beta )/(2* V )