Confused in stat. mech.-thermo., inconsistant result

[fluidistic]

Homework Statement

Hello guys,
Long problem but I'll drastically shorten it. I've a diatomic ideal gas of N molecules (the 2 atoms are distinct). I must calculate the internal energy and the specific heat for high temperatures.
I've got the solution but if I solve it my way in the last step, I get a different result.

Homework Equations

Partition function of one molecule: $Z_1\approx \frac{T}{\omicron}$. This expression is to be found in the solution for this exercise.
Thus, for N molecules, i.e. the whole system: $Z_N\approx \left ( \frac{T}{\omicron} \right ) ^N$ (according to me). These approximations are valid for $T>>\omicron$.
The solution states that from $Z_1$, we can calculate the mean energy per molecule as $\overline{\varepsilon}=-\frac{\partial Z_1 }{\partial \beta}=kT$.

The Attempt at a Solution

My idea was to get the Helmholz free energy and then, from it, getting the internal energy.
I got $A(\beta,N) \approx -\frac{N}{\beta} \ln \left ( \frac{1}{k\beta \omicron} \right )$.
Since $U=\left ( \frac{\partial A}{\partial \beta} \right ) _{V,N}$, I got $U \approx\frac{N}{\beta ^2}[1-\ln (k\beta \omicron)]$. As you can see:
1)My result for U is not extensive (so I'm fried).
2)Dividing my expression for U by N, I don't reach the expression for the average energy per molecule the solution provides.

I don't understand what I did wrong.

 

[fluidistic]

Nevermind guys, I found out the mistake. I had to multiply A by beta before taking the derivative with respect to beta, in order to get the internal energy.