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The inclusive cross sections of prompt J/psi and psi(3686) production are measured at center-of-mass energies from 3.808 to 4.951 GeV. The dataset used is 22 fb(-1) of e(+)e(-) annihilation data collected with the BESIII detector operating at the BEPCII storage ring. The results obtained are in agreement with the previous BESIII measurements of exclusive J/psi and psi(3686) production. The average values obtained for the cross sections measured in the center-of-mass energy ranges from 4.527 to 4.951 GeV for J/psi and from 4.843 to 4.951 GeV for psi(3686), where the impact of known resonances is negligible, are 14.0 +/- 1.7 +/- 3.1 pb and 15.3 +/- 3.0 pb, respectively. For J/psi, the first and the second uncertainties are statistical and systematic, respectively. For psi(3686), the uncertainty is total. These values are useful for testing charmonium production models.