I've never seen the solution method that you have presented so I can't find if there's an error in it. On the other hand if we look at the solution method I know I do not get your book answer either. The fix is trivial and I suspect a typo. I'll run through how I know how to do it and maybe you can compare the two methods to find where your problem is.
I won't bother to prove that your original equation is exact. I'm assuming you already checked that. So I will start with the assumption that there is a function F(x, y) such that
which means that
(The in the integrand is merely to specify we are keeping the value of y fixed during the integration.)
This leads to
We also know that
Integrating this out we get that
So the final solution will be
whereas the book claims . Hence the suspicion of a typo.