Ok, I did those two integrals, then factor out the ##\omega_p## and foil out the c/a operators to get
$$ \frac{1}{2}(a_p a_p^\dagger + a_{-p}a_{-p}^{\dagger}) $$
I see how this is equivalent to the final expression if -p = p, but how do you deal with those negative momenta? Can you just flip...