Evaluate $$I=\int 2e^{\tan^{-1}x+x^2+\ln x}\left(2+\dfrac{2x^3+2x+1}{x+1/x}\right) dx$$
This was an MCQ question, and the solver got the answer as $$I=xe^{\tan^{-1}x+x^2+\ln x}+C$$
However the (apparently) correct answer is $$I=x^2e^{\tan^{-1}x+x^2}+C$$ $$$$The primary discussion was on whether the two functions are equivalent or not, and if not, then which function is the correct antiderivative. At first sight, these two alternatives might appear to be the same, but they are not: $xe^{\tan^{-1}x+x^2+\ln x}+C$ is not defined at $x=0$; $x^2e^{\tan^{-1}x+x^2}+C$ is. I also noticed that the integrand is defined for $x>0$ (and not at $x=0$).
I suspect that somewhere the Fundamental Theorem of Calculus is to be used, however, I am unable to use it correctly.
Note that the domain of the function $ 2e^{\tan^{-1}x+x^2+\ln x}\left(2+\dfrac{2x^3+2x+1}{x+1/x}\right)$ being integrated is $(0, \infty)$.
Therefore, the antiderivative you get $F(x)$ is understood to be defined/restricted on some interval inside $(0, \infty)$.
This makes both answers correct, but if you want to be precise you should emphasize that their domains are $(0, \infty)$.