Measurements are presented of the t-channel single-top-quark production cross section in proton-proton collisions at s√ = 8 TeV. 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 σ t-ch. = 83.6 ± 2.3 (stat.) ± 7.4 (syst.) pb. The single t and t¯ cross sections are measured to be σ t-ch.(t) = 53.8 ± 1.5 (stat.) ± 4.4 (syst.) pb and σ t-ch. (t¯) = 27.6 ± 1.3 (stat.) ± 3.7 (syst.) pb, respectively. The measured ratio of cross sections is R t-ch. = σ t-ch.(t)/σ t-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 V tb is extracted and, in combination with a previous CMS result at s√ = 7 TeV, a value |V tb| = 0.998 ± 0.038 (exp.) ± 0.016 (theo.) is obtained.