  We have measured the spin structure functions $g_1$ and $g_2$ of $^3$He in a
double-spin experiment by inclusively scattering polarized electrons at
energies ranging from 0.862 to 5.07 GeV off a polarized $^3$He target at a
15.5$^{\circ}$ scattering angle. Excitation energies covered the resonance and
the onset of the deep inelastic regions. We have determined for the first time
the $Q^2$ evolution of $\Gamma_1(Q^2)=\int_0^{1} g_1(x,Q^2) dx$,
$\Gamma_2(Q^2)=\int_0^1 g_2(x,Q^2) dx$ and $d_2 (Q^2) = \int_0^1 x^2[
2g_1(x,Q^2) + 3g_2(x,Q^2)] dx$ for the neutron in the range 0.1 GeV$^2$ $\leq
Q^2 \leq $ 0.9 GeV$^2$ with good precision. $ \Gamma_1(Q^2)$ displays a smooth
variation from high to low $Q^2$. The Burkhardt-Cottingham sum rule holds
within uncertainties and $d_2$ is non-zero over the measured range.
