  The photoproduction of $\pi^0$-mesons from the nucleon and deuteron has been
studied for incidents of photon energies up to 1.5 GeV. By using the MAID-2007
model for the process on the nucleon, we predict results for the unpolarized
and helicity-dependent total cross sections of the semi-exclusive reaction
$\gamma d\rightarrow\pi^0X$ ($X=np+d$) with the inclusion of rescattering
effects. We find that rescattering effects yield a substantially large
contribution. The extracted results are compared with the available
experimental data and a satisfactory agreement is obtained. In addition, the
contribution of $\gamma d\rightarrow\pi^0X$ ($X=np+d$) to the finite GDH
integral has been evaluated by explicit integration up to 1.5 GeV and a total
value of 256.96 $\mu$b has been obtained. Convergence of the GDH integral has
been reached.
