Everything was going really well until one week before the exam when the teacher gave us this problem:
$\int \frac{2x^2+13x+19}{x^2+5x+6} dx$
For which I and Wolfram Alpha finds the solution:
$2x+2\ln (x+3) + \ln (x+2) +C$
The thing is that he said that C should be dependent on x in such a way that it is only a constant for those x:s where the answer is defined. Is this really necessary? If the answer isn't defined we won't bother whether the constant is, right?
The antiderivative should include the absolute values, so is $g(x)=2x+\ln |x+3| +\ln |x+2|$ to start. The expression so far is defined at all $x$ other than $-3,-2$. Removing these two points from the reals, there remain three sections, each of which can have an independent constant of integration added to the "antiderivative" $g(x).$
So in this sense the teacher is right, since there can be different constants on the three remaining sections.
What CAS like Wolfram, maple, etc usually give is "an" antiderivative. Then one looks at that result to find the "most general antiderivative", which typically as above may involve breaking the number line into pieces wherein the initial antiderivative is continuous, and adds a separate constant on each piece.
The theorem being used is that, if $f'(x)=0$ holds on an open interval $(a,b)$, and $f(x)$ is continuous on that interval, then $f(x)$ is constant on that interval. So one cannot guarantee the same constant across a discontinuity of the function.