Nice work so far. Something did catch my eye. You have the line
We need to be careful at this point, because the next step is to take an antiderivative with respect to x. Since there are functions of x on the right hand side we cannot do this by putting x's in front of them like we were able to put y in front of F(x) in the previous integration.
Does this make sense? Let me know if there are any questions.