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This is a good example of what i was talking about before in the other thread about equation "form". When you attempt a solution, you go partly by the form the equation is in. If it is in one form, you do it one way, but if it is in a different form, you do it another way.
This equation is not (as is) in variable separation form (simply the variables are not separated) so you can not integrate directly both terms. You can usually only integrate directly when you've first separated the variables (one set on each side of the equation).
Since this equation is in a different form, they are showing you another method you can use when you encounter this form instead of separated variables. This means you go about finding the solution in an entirely or partially different manner. If you try to solve it by integrating both terms as is, you end up with df/dx=4xy instead of 2xy, which would not be correct. If you follow the text, you'll get the right answer.
If you want to try this by integration only, first separate the variables, then integrate both sides. You're lucky here as this equation can be re-arranged with separated variables by dividing both sides by y and then dividing both sides by (x^2-1). Then you can integrate both sides and simplify and get the answer that way.
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