  We obtain a method to compute effective first integrals by combining
Noether's principle with the Kozlov-Kolesnikov integrability theorem. A
sufficient condition for the integrability by quadratures of optimal control
problems with controls taking values on open sets is obtained. We illustrate
our approach on some problems taken from the literature. An alternative proof
of the integrability of the sub-Riemannian nilpotent Lie group of type (2,3,5)
is also given.
