Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
Pleasure. I was taught the habit of using ln A as the constant of integration where it might save having to switch to a different name for the constant part of the solution. But in this case I did end up having to switch from A to another, so using the log constant at first probably didn't help at all.
Anyway, when you say you're looking just for c, well if you mean to identify the constant term in the particular solution, that term is 2/3. Whereas 1/3 is the co-efficient of x^3.
On the other hand, c was the unknown constant in the general solution. (And we used the stated condition to pin it down.)