Let $f$ be a (Riemann) integrable function such that $\displaystyle \int_a^x f\,dx=0$ for all $x\in [a,b]$.
Prove that $\displaystyle \int_a^b fg\,dx=0$ for any integrable $g$.
The question will be easy if we assume that $f$ is continuous, since the given condition will imply $f=0$ everywhere.
However, when $f$ is only required to be integrable, it can be some strange function, like the popcorn function. In this case, I have no idea how to start with, and I can only hope that the given condition will imply $\displaystyle \int_a^b f^2\,dx = 0$.
Then we can use the Cauchy inequality $$\left(\int_a^b fg \,dx\right)^2\leq \left(\int_a^b f^2 \,dx \right) \left( \int_a^b g^2 \,dx\right)=0$$