  Using 2.93~fb$^{-1}$ of data taken at 3.773 GeV with the BESIII detector
operated at the BEPCII collider, we study the semileptonic decays $D^+ \to \bar
K^0e^+\nu_e$ and $D^+ \to \pi^0 e^+\nu_e$. We measure the absolute decay
branching fractions $\mathcal B(D^+ \to \bar K^0e^+\nu_e)=(8.60\pm0.06\pm
0.15)\times10^{-2}$ and $\mathcal B(D^+ \to
\pi^0e^+\nu_e)=(3.63\pm0.08\pm0.05)\times10^{-3}$, where the first
uncertainties are statistical and the second systematic. We also measure the
differential decay rates and study the form factors of these two decays. With
the values of $|V_{cs}|$ and $|V_{cd}|$ from Particle Data Group fits assuming
CKM unitarity, we obtain the values of the form factors at $q^2=0$, $f^K_+(0) =
0.725\pm0.004\pm 0.012$ and $f^{\pi}_+(0) = 0.622\pm0.012\pm 0.003$. Taking
input from recent lattice QCD calculations of these form factors, we determine
values of the CKM matrix elements $|V_{cs}|=0.944 \pm 0.005 \pm 0.015 \pm
0.024$ and $|V_{cd}|=0.210 \pm 0.004 \pm 0.001 \pm 0.009$, where the third
uncertainties are theoretical.
