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.

-Dan