  We calculate alpha'^2-corrections to the entropy of the 5-dimensional
3-charge and the 4-dimensional 4-charge large extremal black holes using the
low energy effective action of the heterotic string theory. In the
4-dimensional case, our results are in agreement with the microscopic
statistical entropy both for the BPS and the non-BPS black holes. In the more
interesting 5-dimensional case, where the direct microscopic stringy
description is still unknown, our results for the BPS black holes are in
agreement with the results obtained from the action supplemented with
R^2-correction obtained by supersymmetric completion of the gravitational
Chern-Simons term. This agreement does not extend to the non-BPS black holes,
for which we propose a different expression for the entropy. We show that the
new expression is supported by certain alpha'^3-order calculations, and by the
arguments based on the AdS/CFT correspondence.
