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.
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