Consider the frame bundle $LM$ of a smooth manifold $M$, and a section $X: U \to LM$ evaluated at each point of $U$ consists of $\{X_{1p}, \ldots, X_{np}\}$ a set of basis for $T_PM$. It is integrable if $[X_i, X_j] = 0$. The claim is for $U$ small enough there exists coordinate charts $(x^1, \ldots, x^n)$ such that $\partial_i = X_i$ if $X$ is integrable.
How do you actually prove it? Just by the definition it is not totally clear that those $X_i$ should be smooth, but it turns out stronger result holds. I know that I am supposed to provide some my own thoughts on it. But I really just do not even know where to start.
Consider $\phi_i$ the local flow of $X_i$. There exists $c>0$ such that ${\phi_i}_t$ is defined if $t\in [-c,c]$. Define, the coordinate at $x$ by $(t_1,...,t_n)={\phi_1}_{t_1}\circ {\phi_2}_{t_2}\circ...\circ {\phi_n}_{t_n}$.