If $f$ and $f'$ are integrable, then $f'$ has integral $0$ I have the problem

Prove that if $f$  and $f'$ are Lebesgue integrable over $\mathbb{R}$, then $\int_\mathbb{R}f' = 0$, where $f'$ is defined everywhere.

Honestly, not sure where to start. I had the idea to perhaps use the fundamental theorem of calculus $f(x) = \int_0^x f' + f(0)$ and take limits as $x$ goes to $0$ and trying to work from there. However, I'm not sure that the theorem holds in this case. Any help getting started is appreciated!

As a counter-example you can take a compactly supported function, say $f(x)=x^2$ over $[0, 1]$.
Edit: As pointed out in the comments, the answer to this question really depends on how you interpret the question. This answer assumes that $f'$ means the derivative of the $L^1$ function $f$ which is defined almost everywhere.
