There is at least one mistake per line, both algebra and calculus errors. Try this:
So far, I haven't done any calculus whatsoever, and thus there should be no constant of integration yet. In addition, the title of your post indicated that you are taking an antiderivative. It puzzled me, then, when I saw the notation g'(x), which is the notation for a derivative.
In any case, you can now integrate term-by-term. What do you get?