Consider $$AC^{n}:=\{f\in{AC}:f^{(k)}\in{AC}, 1\leq k \leq n-1\}$$ where $AC$ stands for the space of absolutely continuous functions.
Now, let $f,g\in{L_{loc}^{1}(a,b)}$ and $$\int_{a}^{b}g(x)\phi(x)dx=(-1)^{n} \int_{a}^{b}f(x)\phi^{(n)}(x)dx$$ for all $\phi\in C_{0}^{\infty}(a,b)$. Show that $f\in{AC_{loc}^{n}(a,b)}$ and $g(x)=f^{(n)}(x)$ a.e. on $(a,b)$.