= Solution
Assume <thermal equilibrium>, nondegenerate carriers, and isotropic <parabolic band dispersions>, with the supplied <effective masses> interpreted as density-of-states masses. Take one effective valley with two <electron spin> states for each band; any additional valley multiplicity must instead be absorbed into the masses or explicit degeneracy factors.
Let the <conduction band> edge be $E_c$, the <valence band> edge $E_v$, and $E_g=E_c-E_v$. The spin-summed conduction <density of states> per <volume> is
$$
g_c(E)=\frac1{2\pi^2}\left(\frac{2m_e^*}{\hbar^2}\right)^{3/2}\sqrt{E-E_c},\qquad E\geq E_c.
$$
Using the nondegenerate limit of the <Fermi-Dirac distribution> and the substitution $x=(E-E_c)/(k_BT)$ gives
$$
n=\int_{E_c}^\infty g_c(E)e^{-(E-\mu)/(k_BT)}dE
=N_c e^{-(E_c-\mu)/(k_BT)},
\quad N_c=2\left(\frac{m_e^*k_BT}{2\pi\hbar^2}\right)^{3/2}.
$$
The factor $\sqrt\pi/2$ in the supplied integral produces the normalization. Counting missing <valence band> <electrons> similarly gives
$$
p=N_v e^{-(\mu-E_v)/(k_BT)},\qquad
N_v=2\left(\frac{m_h^*k_BT}{2\pi\hbar^2}\right)^{3/2}.
$$
Multiplication cancels the <chemical potential>, establishing the <semiconductor mass-action law>:
$$
\boxed{np=N_cN_v e^{-E_g/(k_BT)}
=4\left(\frac{k_BT}{2\pi\hbar^2}\right)^3(m_e^*m_h^*)^{3/2}e^{-E_g/(k_BT)}.}
$$
The result does not require an <intrinsic semiconductor>: <semiconductor doping> changes $n$ and $p$ separately while preserving this product under the stated equilibrium and nondegeneracy assumptions.
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