step 1:
x = integral(from 0 to v) dv/(z^2-v^2)
step 2:
x = 1/2z ln((q+v)/(q-v))
How do you get from step 1 to step 2 ?
The indefinite integral of
dv/(z^2-v^2),
with z being a constant, is
[1/(2z)]log[(z+v)/(z-v)]
Evaluate that at v=v' and subtract the value for v=0, to get the definite integral.
The method of partial fractions can used to get the integral. It involves rewriting 1/[(z^2-v^2} as
[1/(2z)][1/(z+v) - 1/(z-v)]