Problem 1.

Clear[f]
$Assumptions = nu > 0 ;
f[nu_, z_] = Integrate[ x^(nu -1)/(E^x/z - 1), {x, 0, Infinity}]

Module[{w},
w = {1/2, 1, 3/2} ;
Plot[
Evaluate[Map[ f[#, z] &, w]],
{z, 0, 1},
PlotRange -> {0, 10},

PlotLegends->Placed[Map[ (Subscript["(SuperMinus[f])",  Text[#]]) Text["(z)"]&, w],{Left, Top}]]
]
Plot[
f[3/2, z],
{z, 0, 1},
PlotRange -> {0, 4},
AxesLabel->{z, "(Subscript[(SuperMinus[f]), 3/2])" Text["(z)"]}
]

numerical calculations for problem set 7, p2.

Clear[t, kb, hbar]
t[n_, m_, kb_, hbar_] = (1/kb) (n/Zeta[3/2])^(2/3) 2 Pi hbar^2/m
hbar=1.05457173×10^(-34)  (* m^2 kg/s *)
kb = 1.3806488×10^(-23) (* m^2 kg s^-2 *)
t[n, m, kb, hbar]
t[1.88 10^28, 6.64 10^(-27), kb, hbar]
t[10^24, 1.443 10^(-25), kb, hbar]
Zeta[3/2] // N

Problem 3.

Clear[i]
i = Integrate[ x^2/(E^x-1), {x, 0, Infinity}]
i // N
Plot[x^2/(E^x-1),{x, 0, 10}, PlotRange -> Full]