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I think they're just being very sloppy with their definitions of the arbitrary functions f and g. In that if you integrate some arbitrary function, then you get some other arbitrary function - and they just used the same name. That's my best guess for what they're doing.

Actually, there's a whole lot about that 'answer' that I don't like. I assume g(x) becoming g(y) is just a mistake. The implicit redefinition of the functions is bogus. And even if they were given new letters, I don't like that the integration constants / functions are introduced before the integral is done. Like, I guess they are a result of integrating the LHS - but then we're implicitly assuming that the other integral will give a constant of zero... To me that doesn't look like good technique. But then again, maybe I've misunderstood the whole thing!