Making the units of a analytically calculated PSD consistent with the units of an FFT
You do this by doing two things:

First establish what the relationship of scale exists between your A/D converter and the numerical values that go into your DFT.  Find out how many volts corresponds to a value of $1.0$ in the samples that go into the DFT.

The next thing you gotta do is express your integral above without that $\frac{1}{T}$ factor (which is an error) as a Riemann summation having equal-width little rectangles of width equal to the sampling period.  That will become identical to the summation of the DFT.


That should spell it out.