  Vela X is a pulsar wind nebula in which two relativistic particle populations
with distinct spatial and spectral distributions dominate the emission at
different wavelengths. An extended $2^\circ \times 3^\circ$ nebula is seen in
radio and GeV gamma rays. An elongated cocoon prevails in X-rays and TeV gamma
rays. We use 9.5 years of data from the Fermi Large Area Telescope (LAT) to
disentangle gamma-ray emission from the two components in the energy range from
10 GeV to 2 TeV, bridging the gap between previous measurements at GeV and TeV
energies. We determine the morphology of emission associated to Vela X
separately at energies < 100 GeV and > 100 GeV, and compare it to the
morphology seen at other wavelengths. Then, we derive the spectral energy
distribution of the two gamma-ray components over the full energy range. The
best fit to the LAT data is provided by the combination of the two components
derived at energies < 100 GeV and > 100 GeV. The first component has a soft
spectrum, spectral index $2.19\pm0.16^{+0.05}_{-0.22}$, and extends over a
region of radius $1.36^\circ\pm0.04^\circ$, consistent with the radio nebula.
The second component has a harder spectrum, spectral index
$0.9\pm0.3^{+0.3}_{-0.1}$, and is concentrated over an area of radius
$0.63^\circ\pm0.03^\circ$, coincident with the X-ray cocoon that had already
been established to account for the bulk of the emission at TeV energies. The
spectrum measured for the low-energy component corroborates previous evidence
for a roll-over of the electron spectrum at energies of a few tens of GeV
possibly due to diffusive escape. The high-energy component has a very hard
spectrum: if the emission is produced by electrons with a power-law spectrum
the electrons must be uncooled, and there is a hint that their spectrum may be
harder than predictions by standard models of Fermi acceleration at
relativistic shocks. (Abridged)
