  We develop the calculation of the divergent part of one-loop covariant
effective action for scalar fields minimally and non-minimally coupled to
gravity using the generalized Schwinger-DeWitt technique. We derive the
field-space metric using Vilkovisky's prescription and obtain new terms in the
one-loop corrections which are absent in past literatures with trivial choices
of field-space metric. We obtain the covariant versions of past non-covariant
results by using Barvinsky and Vilkovisky's extension to the ordinary
Schwinger-DeWitt approach. For completeness, we study the one-loop
gravitational corrections for a dilaton-extended two-field Starobinsky model
and compare with known results.
