I'm really triggered by not finding the mistake in the following lines.
Suppose $f \in C^\infty_0$ ($C_0$ stands for a function with compact support). Then,
$ \int_{-\infty}^{+\infty}f'(x) = [f(x)]^{+\infty}_{-\infty} $ = 0.
I sense that my anti-derivative is wrong, but I can't explain to myself why. Thanks alot for your help.
No mistake.
Look at the following graphics representing the case of an even function $f$ (red) ; its derivative (blue) will be odd and the positive and the negative area parts will compensate each other :