= Solution
Assume nondegenerate three-dimensional parabolic band edges, a fixed gap and approximately temperature-independent density-of-states effective masses. Let $\mu$ be the <chemical potential>, $E_c$ and $E_v$ the band edges, and $N_c,N_v$ their <effective density of states>. Boltzmann occupation gives
$$
n=N_c e^{-(E_c-\mu)/(k_BT)},\qquad
p=N_v e^{-(\mu-E_v)/(k_BT)},\qquad E_g=E_c-E_v.
$$
For one spin-degenerate spherical conduction extremum, direct integration of $g_c(E)\propto\sqrt{E-E_c}$ gives
$$
N_c=2\left(\frac{m_c^*k_BT}{2\pi\hbar^2}\right)^{3/2},
$$
and similarly for the <valence band> with its appropriate density-of-states <mass>. Valley and additional band degeneracies can be included in these prefactors. The $T^{3/2}$ factor follows from the integral $\int_0^\infty\sqrt\epsilon\,e^{-\epsilon/(k_BT)}d\epsilon\propto T^{3/2}$.
Multiplication removes the <chemical potential>:
$$
np=N_cN_v e^{-E_g/(k_BT)}.
$$
The <semiconductor mass-action law> combined with intrinsic <charge neutrality>, $n=p$, therefore gives
$$
\boxed{n_i=\sqrt{N_cN_v}\,e^{-E_g/(2k_BT)}=AT^{3/2}e^{-E_g/(2k_BT)}}.
$$
The intrinsic <chemical potential> is not necessarily exactly midgap: $\mu_i=(E_c+E_v)/2+(k_BT/2)\ln(N_v/N_c)$. Equal effective masses are unnecessary for the displayed carrier-density law. A changing gap, effective <mass> or degeneracy would make the prefactor or activation law <temperature> dependent.
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