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Measurements are presented of the t-channel single-top-quark production cross section in proton-proton collisions at √s = 8TeV. The results are based on a data sample corresponding to an integrated luminosity of 19.7 fb−1 recorded with the CMS detector at the LHC. The cross section is measured inclusively, as well as separately for top (t) and antitop (t¯), in final states with a muon or an electron. The measured inclusive t - channel cross section is σt-ch. = 83.6 ± 2.3 (stat.) ± 7.4 (syst.) pb. The single t and t ¯ cross sections are measured to be σt-ch.(t) = 53.8 ± 1.5 (stat.) ± 4.4 (syst.) pb and σt-ch.(t¯) = 27.6 ± 1.3 (stat.) ± 3.7 (syst.) pb, respectively. The measured ratio of cross sections is Rt-ch. = σt-ch.(t)/σt-ch.(t¯) = 1.95±0.10 (stat.)±0.19 (syst.), in agreement with the standard model prediction. The modulus of the Cabibbo-Kobayashi-Maskawa matrix element Vtbis extracted and, in combination with a previous CMS result at √s = 7TeV, a value |Vtb| = 0.998 ± 0.038 (exp.) ± 0.016 (theo.) is obtained.