A "crazy" way to get Z(T,V,N) for classical ideal gas (Declaration: I think it will work, but I haven't tried it out) You will now use Eq. (3)
Z(T,V,N)=∫0[infinity]W(E,V,N)e−βEdE
to calculate Z(T,V,N) for a classical ideal gas, where W(E,V,N)dE is the number of N-particle (micro)states with energies in the interval E to E+dE. Once upon a time, we did the classical ideal gas by counting the microstates and found
W<(E,V,N)=N!h3NVN(23N)!(2πmE)3N/2
for the number of microstates with energy less than or equal to E. It is related to W(E,V,N) by W(E,V,N)=∂W<(E,V,N)/∂E Your action: Evaluate the integral in Eq. (3) and show that the result is the classical ideal gas partition function in Eq. (6) [Remark: This method may sound "crazy" because when we know W(E,V,N), we could use S=klnW to extract all the physics without evaluating Z at all. But physics is physics, we simply follow a version of the definition and obtain the expected Z for a classical ideal gas. It is quite nice after all!]