The production cross-sections of $J/psi$ mesons in proton-proton collisions at a centre-of-mass energy of $sqrt{s}=5$ TeV are measured using a data sample corresponding to an integrated luminosity of $9.13pm0.18~text{pb}^{-1}$, collected by the LHCb experiment. The cross-sections are measured differentially as a function of transverse momentum, $p_{text{T}}$, and rapidity, $y$, and separately for $J/psi$ mesons produced promptly and from beauty hadron decays (nonprompt). With the assumption of unpolarised $J/psi$ mesons, the production cross-sections integrated over the kinematic range $0<p_{text{T}}<20~text{GeV}/c$ and $2.0<y<4.5$ are $8.154pm0.010pm0.283~mutext{b}$ for prompt $J/psi$ mesons and $0.820pm0.003pm0.034~mutext{b}$ for nonprompt $J/psi$ mesons, where the first uncertainties are statistical and the second systematic. These cross-sections are compared with those at $sqrt{s}=8$ TeV and $13$ TeV, and are used to update the measurement of the nuclear modification factor in proton-lead collisions for $J/psi$ mesons at a centre-of-mass energy per nucleon pair of $sqrt{s_{text{NN}}}=5$ TeV. The results are compared with theoretical predictions.
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