I am reading the proof of the Euler Lagrange Equation, namely, let $X \subset \mathbb{R}^d$ be an open set and $L\in C^2(X\times \mathbb{R}^d;\mathbb R)$, and $\phi \in C^1([0,T];X)$. Let $\xi \in C^1([0,T]; \mathbb{R}^d)$ such that $\xi(0)=0,\xi(T)=0$.
Then the proof shows that we have $$\int_0^T (-\partial_t[\partial_{\dot{\phi}}L(\phi,\dot{\phi})]+\partial_\phi L(\phi,\dot{\phi}))\xi(t)dt=0$$ for all such $\xi$.
Thus we have $$-\partial_t[\partial_{\dot{\phi}}L(\phi,\dot{\phi})]+\partial_\phi L(\phi,\dot{\phi})=0.$$
I believe this is a result of some form of vanishing theorem for integrals, but I cannot find a way to prove this statement. Why is the above $0$ if its integral multiplied with $\xi$ for all $C^1$ functions with starting and end points at $0$, equal to $0$?