= Solution
The classical one-particle <partition function>, with phase-space cell $h^3$, is
$$
Z_1=\frac1{h^3}\int d^3p\,d^3x\,
e^{-[p^2/(2m)+U(x)]/(kT)}
=\left(\frac{2\pi mkT}{h^2}\right)^{3/2}
\int e^{-U/(kT)}\,d^3x.
$$
For the homogeneous trap, rescale $r=V^{1/3}(kT)^{1/(2n)}u$. The spatial measure contributes $V(kT)^{3/(2n)}$, so
$$
\boxed{Z_1=VI_n(kT)^a,\qquad
a=\frac32+\frac3{2n}=\frac{3(n+1)}{2n}}.
$$
Here $I_n=(2m\pi/h^2)^{3/2}\int_0^\infty4\pi u^2e^{-u^{2n}}\,du$, as specified; the last integral is $2\pi\Gamma(3/(2n))/n$.
Using the printed convention $Z_N=Z_1^N$, the <free energy> is
$$
\boxed{F=-NkT[\log V+a\log(kT)+\log I_n]}.
$$
This convention treats the particles as labeled. For indistinguishable classical particles, $Z_N=Z_1^N/N!$ adds $kT\log N!$ to $F$. It changes the <entropy> by $-k\log N!$ but leaves the following force, <energy> and <heat capacities> unchanged. This normalization qualification is required to distinguish the printed expression from the usual Gibbs-corrected gas <free energy>.
The generalized force conjugate to the external parameter $V$ is $P=-(\partial F/\partial V)_T$, hence
$$
\boxed{P=\frac{NkT}{V},\qquad PV=NkT}.
$$
It is the same equation of state as a hard-wall ideal gas, although $V$ now dilates a soft confining potential rather than being the occupied geometric volume.
Writing $L=\log V+a\log(kT)+\log I_n$, thermodynamic differentiation gives
$$
\boxed{S=Nk(L+a),\qquad E=F+TS=aNkT,\qquad C_V=aNk}.
$$
The mean <kinetic energy> is $3NkT/2$ and the trap potential contributes $3NkT/(2n)$. At fixed $P$, $V=NkT/P$, so $\partial_TL|_P=(a+1)/T$. Consequently
$$
\boxed{C_P=T(\partial S/\partial T)_P=(a+1)Nk=C_V+Nk}.
$$
The same result follows from <enthalpy> $E+PV=(a+1)NkT$. These identities describe the <classical ideal gas in a homogeneous soft trap>.
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