Production cross-sections of prompt charm mesons are measured using data from $pp$ collisions at the LHC at a centre-of-mass energy of $5,$TeV. The data sample corresponds to an integrated luminosity of $8.60pm0.33,$pb$^{-1}$ collected by the LHCb experiment. The production cross-sections of $D^0$, $D^+$, $D_s^+$, and $D^{*+}$ mesons are measured in bins of charm meson transverse momentum, $p_{text{T}}$, and rapidity, $y$. They cover the rapidity range $2.0<y<4.5$ and transverse momentum ranges $0 < p_{text{T}} < 10, text{GeV}/c$ for $D^0$ and $D^+$ and $1 < p_{text{T}} < 10, text{GeV}/c$ for $D_s^+$ and $D^{*+}$ mesons. The inclusive cross-sections for the four mesons, including charge-conjugate states, within the range of $1 < p_{text{T}} < 8, text{GeV}/c$ are determined to be sigma(pprightarrow D^0 X) = 1004 pm 3 pm 54,mutext{b} sigma(pprightarrow D^+ X) = 402 pm 2 pm 30,mutext{b} sigma(pprightarrow D_s^+ X) = 170 pm 4 pm 16,mutext{b} sigma(pprightarrow D^{*+} X)= 421 pm 5 pm 36,mutext{b} where the uncertainties are statistical and systematic, respectively.
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