The semiclassical exchange approximation and three free-electron-gas exchange approximations are applied to electron scattering from the hydrogen atom. In the triplet spin state the results are similar to our previous findings [M. E. Riley and D. G. Truhlar, J. Chem. Phys. 63, 2182 (1975)] for electron scattering by closed-shell atoms. However these results, already of useful accuracy, are improved by the inclusion of one-electron exchange terms or an orthogonality constraint. For the singlet spin state it is essential to include the one-electron exchange terms except at high impact energies (greater than about 50 eV). The approximation of Lippmann, Schey, Burke, and Chandra for treating the triplet spin state and exchange with closed-shell targets is tested and found to be less accurate than the present approximations.