While reading the proof of Lemma 2 in the following link, I realized they only proved the case of a nonnegative function $f$, but that's not an hypothesis of the lemma. So, what happens if $f$ takes negative values? Does the lemma remain true? Is the proof similar to the nonnegative case?
Lemma 2: Let $f$ is a Lebesgue integrable function on $[a, b]$ and let $F(x)=\int_a^x f(t) d t$. If for all $x \in[a, b]$ we have that $F(x)=0$ then $f(x)=0$ almost everywhere on $[a, b]$
Link of the proof: http://mathonline.wikidot.com/integral-criteria-for-functions-to-be-zero-almost-everywhere
Following the proof of the Lemma 2 in the link it shows that $m(\{x\in[a,b]:f(x)>0\})=0$, therefore $f$ is non-positive almost everywhere. Then it follows from the Lemma 1 that $-f=0$ almost everywhere, so we conclude that $f=0$ a.e.
That is, if we set $S:=\{x\in[a,b]: f(x)>0\}$ then $$ \int_{[a,b]}f\mathop{}\!d \lambda =\overbrace{\int_{S}f\mathop{}\!d \lambda}^{=0} +\int_{[a,b]\setminus S}f \mathop{}\!d \lambda =-\int_{[a,b]\setminus S}|f|\mathop{}\!d \lambda =0\\[2ex] \therefore\quad \mathbf{1}_{[a,b]\setminus S}\,|f|=0\text{ a.e. }\implies \mathbf{1}_{[a,b]\setminus S}\,f=0\text{ a.e. }\implies f=0\text{ a.e. } $$