  Working in the basis where the charged-lepton Yukawa matrix is diagonal and
making the $\tau$-dominance approximations, we analytically derive integral
solutions to the one-loop renormalization-group equations (RGEs) for neutrino
masses, flavor mixing angles, CP-violating phases and the Jarlskog invariant
under the standard parametrization of the PMNS matrix in the standard model or
its minimal supersymmetric extension for both Majorana and Dirac neutrinos.
With these integral solutions, we carry out numerical calculations to
investigate the RGE running of lepton flavor mixing parameters and the Jarlskog
invariant, and also compare these integral solutions with the exact results
obtained by numerically solving the one-loop RGEs. It is shown that these
integral solutions coincide with the exact results and can well describe the
evolution of lepton flavor mixing parameters and the Jarlskog invariant in most
cases. Some important features of our integral solutions and the evolution
behaviors of relevant flavor parameters are also discussed in detail both
analytically and numerically.
