$f,g$ are continuous function $[a,b]$.Suppose $$\int_a^b{f(t)h(t)+g(t)h'(t)}dt = 0$$ for every $h$ belonging to $C^1[a,b]$ with $h'(a)=h'(b)=0$. Why it is true that
(1)$\int_a^bf(t)dt=0$---->This is clear.
(2)$g$ belongs to $C^1[a,b]$ and $g'=f$
I have no idea how to solve the second problem.
Define $v:[a,b]\rightarrow\mathbb{R}$ by $v(x)=\int_a^x f(t)dt$. By using Fubini's theorem prove that $$\int_a^b vh'=-\int_a^b fh,\ \forall\ h\in C_c^1((a,b))$$
This implies that $$\int_a^b (v-g)h'=0,\ \forall\ h\in C_c^1((a,b)) $$
As @Olod mentioned in the comments, by using du Bois-Reymond's lemma, you can prove that $v-g$ is constant. Another book where you can find the proof is Brezis, in pages 204-206.