= Solution
In the very early universe, assume species $i$ is in <thermal equilibrium>, is <ultrarelativistic> so that $kT\gg m_ic^2$ and $E(p)\simeq pc$, and has negligible <chemical potential>, $\mu_i\simeq0$. Its occupation factor is then the <Bose-Einstein distribution> for a boson or the <Fermi-Dirac distribution> for a fermion.
Substitute $E=pc$ and set $y=pc/(kT)$. Since $p=(kT/c)y$ and $dp=(kT/c)\,dy$,
$$
\begin{aligned}
\epsilon_i
&=\frac{4\pi g_i}{h^3}\int_0^\infty
\frac{pc\,p^2\,dp}{e^{pc/(kT)}\mp1}\\
&=\boxed{\frac{4\pi g_i}{(hc)^3}(kT)^4
\int_0^\infty\frac{y^3}{e^y\mp1}\,dy}.
\end{aligned}
$$
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