How do I calculate the velocity dispersion of a galaxy given an integral field spectrograph? I couldn't find on the web how to get the velocity dispersion and velocity maps of galaxies from the 3D data cube that I get from the integral field spectrograph.
If the 3D data cube is sufficiently highly resolved, try to find Doppler shifts in the spectra by looking for typical absorption/emission lines. Doppler shifts correspond to radial velocities. Calculate the mean radial velocity, and e.g. the standard deviation of the radial velocities, as velocity dispersion.

The velocity dispersion of galaxies leads to a widening of spectral lines in comparison to a reference star. A couple of methods are discussed in this paper, mostly some kind of Fourier fitting, but direct fitting is used, too.

Redshifts may contain a gravitational component.

There exist data reduction software tools.

A paper describing data reduction and velocity analysis using the XOASIS tool.