The neutron longitudinal and transverse asymmetries A1n​ and A2n​ have been extracted from deep inelastic scattering of polarized electrons by a polarized 3He target at incident energies of 19.42, 22.66 and 25.51 GeV. The measurement allows for the determination of the neutron spin structure functions g1n​(x,Q2) and g2n​(x,Q2) over the range 0.03<x<0.6 at an average Q2 of 2 (GeV/c)2. The data are used for the evaluation of the Ellis-Jaffe and Bjorken sum rules. The neutron spin structure function g1n​(x,Q2) is small and negative within the range of our measurement, yielding an integral ∫0.030.6​g1n​(x)dx=−0.028±0.006(stat)±0.006(syst). Assuming Regge behavior at low x, we extract Γ1n​=∫01​g1n​(x)dx=−0.031±0.006(stat)±0.009(syst). Combined with previous proton integral results from SLAC experiment E143, we find Γ1p​−evaluatedusing\alpha_s = 0.32\pm 0.05$