Statistical mechanics - average particle energy, average kinetic energy

Question

I'm looking at derivations for average particle energy giving $E=kT$:

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/bolapp.html

And average particle kinetic energy giving $K_E=\dfrac{3}{2}kT$:

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/molke.html#c1

The concept of potential energy doesn't enter the above derivations.

I'm confused by the difference in the two situations. Specifically in the first derivation, the average value is obtained by integrating over $dE$, whereas the second is obtained by integrated over $dv$. Is there a reason for the difference?

Basically I don't understand why the first derivation doesn't equally apply to the second situation, leading to an average kinetic energy of $kT$.