= Solution
{c}
Let $\beta=1/(k_BT)$, with $k_B$ the <Boltzmann constant>, $T$ the absolute <temperature>, and $\mu$ the common equilibrium <chemical potential>. The <conduction band> minimum and <valence band> maximum are $\epsilon_c$ and $\epsilon_v$; their separation is the <band gap> $E_g=\epsilon_c-\epsilon_v$. The positive parameters $m_e,m_h$ in the supplied <density of states> functions are the <band effective masses> of <electrons> and <electron holes>, not necessarily the bare <electron mass>.
The <electron> occupation is the <Fermi-Dirac distribution> $f(\epsilon)=[e^{\beta(\epsilon-\mu)}+1]^{-1}$. The appropriate statistical factors are $f_c=f$ for <electrons> and $f_v=1-f$ for <electron holes>. In a <nondegenerate semiconductor>, $\epsilon_c-\mu\gg k_BT$ and $\mu-\epsilon_v\gg k_BT$, so the <Boltzmann approximation> gives
$$
f_c(\epsilon)\simeq e^{-\beta(\epsilon-\mu)},\qquad
f_v(\epsilon)\simeq e^{\beta(\epsilon-\mu)}.
$$
For the <conduction band>, set $u=\epsilon-\epsilon_c$ and then $\xi=\beta u$. The supplied integral yields
$$
\begin{aligned}
n&=\frac{(2m_e)^{3/2}}{2\pi^2\hbar^3}
e^{-\beta(\epsilon_c-\mu)}\int_0^\infty u^{1/2}e^{-\beta u}\,du\\
&=2\left(\frac{m_e}{2\pi\beta\hbar^2}\right)^{3/2}
e^{-\beta(\epsilon_c-\mu)}=N_c e^{-\beta(\epsilon_c-\mu)}.
\end{aligned}
$$
For the <valence band>, set $u=\epsilon_v-\epsilon$ to obtain
$$
p=N_v e^{-\beta(\mu-\epsilon_v)},\qquad
N_v=2\left(\frac{m_h}{2\pi\beta\hbar^2}\right)^{3/2}.
$$
The <effective density of states> factors $N_c,N_v$ include the two <spin> states already present in the supplied <density of states> functions. Multiplication cancels the <chemical potential> and gives the <semiconductor mass-action law>
$$
\boxed{np=N_cN_v e^{-\beta E_g}
=4\left(\frac{\sqrt{m_em_h}\,k_BT}{2\pi\hbar^2}\right)^3
e^{-E_g/(k_BT)}=n_i^2.}
$$
Here $n_i=\sqrt{N_cN_v}e^{-E_g/(2k_BT)}$ is the <intrinsic carrier concentration>. Equality $n=p=n_i$ additionally requires an electrically neutral <intrinsic semiconductor>. The product law also applies to an equilibrium doped <semiconductor> while both populations remain nondegenerate. The derivation assumes parabolic <energy bands>, the stated <spin degeneracy>, and a single common <chemical potential>.
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