On the right hand side, the integration is simple, with A being the constant of integration, so I imagine you're asking about what they did on the left hand side.
First, they used the fact that to get
Next, you use the method of partial fractions:
for some A, B.
To find A and B, we multiply this by to get:
Equating the coefficients of powers of v, we get and
This gives and . Thus:
Now, note that , so ,
Then you use the fact that to get
which is the left hand side of their result.