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u = 3/5 N Subscript[\[Epsilon], F]

Now I know total energy of N particles is this integral

u = \!\(

\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(\[Infinity]\)]\(\

\[Epsilon]\ P[\[Epsilon]] \[DifferentialD]\[Epsilon]\)\)

which is made up of the density of the states and probability of the electron to occupy level with energy \[Epsilon] at temp T.

So P[\[Epsilon]] is this big horrible looking thing. My guess is that there must be a be an easy way to integrate it that comes about from absolute zero tempature because the final answer seems so nice.

Any help, would be nice. thanks!