  Coupled-channel effects are taken into account for the study of scalar form
factors in semi-leptonic $D\to\pi\bar{\ell}\nu_\ell$ and $D\to
\bar{K}\bar{\ell}\nu_\ell$ decays, by solving the Muskhelishvili-Omn\`es
integral equations. As inputs, we employ the unitarized amplitudes taken from
chiral effective theory for the region not far away from thresholds, while, at
higher energies of the Goldstone bosons, proper asymptotic conditions are
employed. Within Muskhelishvili-Omn\`es formalism, the scalar form factors are
represented in terms of Omn\`es matrix multiplied by a vector of polynomials.
We reduce the number of subtraction constants by matching to the scalar form
factors derived in chiral perturbation theory up to next-to-leading order. The
recent lattice QCD data by ETM collaboration for ${D\to\pi}$ and
${D\to\bar{K}}$ scalar form factors are simultaneously well described. The
scalar form factors for $D\to\eta$, $D_s\to \bar{K}$ and $D_s\to \eta$
transitions are predicted in their whole kinematical regions. Using our fitting
parameters, we also extract the following Cabibbo-Kobayashi-Maskawa elements:
$|V_{cd}|=0.243(_{-12}^{+11})_{\rm sta.}({3})_{\rm sys.}(3)_{\rm exp.}$ and
$|V_{cs}|=0.950(_{-40}^{+39})_{\rm sta.}(1)_{\rm sys.}(7)_{\rm exp.}$. The
approach used in this work can be straightforwardly extended to the
semileptonic $B$ decays.
