  The analysis of signals in directional dark matter (DM) detectors typically
assumes that the directions of nuclear recoils can be measured in the Galactic
rest frame. However, this is not possible with all directional detection
technologies. In nuclear emulsions, for example, the recoil events must be
detected and measured after the exposure time of the experiment. Unless the
entire detector is mounted and rotated with the sidereal day, the recoils
cannot be reoriented in the Galactic rest frame. We examine the effect of this
`time integration' on the primary goals of directional detection, namely: (1)
confirming that the recoils are anisotropic; (2) measuring the median recoil
direction to confirm their Galactic origin; and (3) probing below the neutrino
floor. We show that after time integration the DM recoil distribution retains a
preferred direction and is distinct from that of Solar neutrino-induced
recoils. Many of the advantages of directional detection are therefore
preserved and it is not crucial to mount and rotate the detector. Rejecting
isotropic backgrounds requires a factor of 2 more signal events compared with
an experiment with event time information, whereas a factor of 1.5-3 more
events are needed to measure a median direction in agreement with the
expectation for DM. We also find that there is still effectively no neutrino
floor in a time-integrated directional experiment. However to reach a cross
section an order of magnitude below the floor, a factor of 8 larger exposure is
required than with a conventional directional experiment. We also examine how
the sensitivity is affected for detectors with only 2D recoil track readout,
and/or no head-tail measurement. As for non-time-integrated experiments, 2D
readout is not a major disadvantage, though a lack of head-tail sensitivity is.
