Confusion to put the constant of integration

Question

If $F(x) = x^2+C$, where $C$ is the constant of integration, is it valid to write, $$F(x) = \int 2x \, \mathrm{d}x$$ without putting the $+\,C$ once more in the left hand side, or do I really have to write $$F(x) + C = \int 2x \, \mathrm{d}x$$ despite the fact that $+\,C$ is already specified in $F(x)$?

Answer 1

Both ways are valid, but the first way is used more often. The reason that $C$ is there is to indicate that a whole set of functions, each differing from another by only the value of $C$, satisfies the integral.