  We show that the large corrections due to final state interactions (FSI) in
the D^+\to \pi^-\pi^+\pi^+, D^+_s\to \pi^-\pi^+\pi^+, and D^+\to K^-\pi^+\pi^+
decays can be accounted for by invoking scattering amplitudes in agreement with
those derived from phase shifts studies. In this way, broad/overlapping
resonances in S-waves are properly treated and the phase motions of the
transition amplitudes are driven by the corresponding scattering matrix
elements determined in many other experiments. This is an important step
forward in resolving the puzzle of the FSI in these decays. We also discuss why
the \sigma and \kappa resonances, hardly visible in scattering experiments, are
much more prominent and clearly visible in these decays without destroying the
agreement with the experimental \pi\pi and K\pi low energy S-wave phase shifts.
