I know thats the following statement is true.
$f,g$ are continuous function $[a,b]$.Suppose $\int\limits_a^bf(t)h(t)+g(t)h'(t) \, dt=0$ for every $h$ belonging to $C_0^{\infty}[a,b]$, then $g$ is differentiable and $\dot{g}(t)=f(t),\text{ }t\in[a,b] \text{ a.e.}$
Is the theorem also true for functions $f,g \in L^1[a,b]$ and can you give me a reference for it?
Let $g, f \in L^1[a,b]$, and suppose that $\nu$ is a weak derivative for $g$. That means $$\int_a^b g h' dx = - \int_a^b \nu h dx$$ for all $h \in C^\infty_0[a,b]$.
If $\int_a^b fh + gh' dx = 0$ for all $h \in C^\infty_0$, then $$\int_a^b h fdx = \int_a^b h \nu dx.$$
Therefore when viewed as measures $f$ and $\nu$ agree after integration against $C^\infty_0$ functions. Which means that $f = \nu$ almost everywhere. This comes from measure theory.