The antiderivative of zero will be any function whose derivative is zero. So, as your book says, any constant function will be an antiderivative of zero.
The notation $\int f(x) dx$ means nothing else but "an antiderivative of $f(x)$".
The example you mention, in particular, shows that you have to careful with your arithmetic manipulations: when you take the constant zero "out" in $\int 0\,dx$, you would get $0\,\int 1\,dx=0$, as Mario Carneiro says. But if you add a constant to an antiderivative, you still get an antiderivative, so any equality between antiderivatives is up to a constant.
So, if you show that $\int f(x)\,dx=0$, what you know that is $\int f(x)\,dx$ differs from $0$ by a constant: so $\int f(x)\,dx=c$ for some constant.