= Solution
Let $\pi:T^*X\to X$ be the <cotangent bundle> projection and $\lambda$ its <canonical one-form on a cotangent bundle>, defined by $\lambda_{(x,\xi)}(v)=\xi(d\pi(v))$. Use the <symplectic form> $\omega=-d\lambda$, so in local coordinates $\omega=\sum_jdx_j\wedge d\xi_j$.
The graph map $s_\beta(x)=(x,\beta_x)$ is a <smooth embedding>, with $s_\beta^*\lambda=\beta$, and hence
$$
s_\beta^*\omega=-d\beta.
$$
Its image has dimension $\dim X$, half the dimension of $T^*X$. By the definition of a <Lagrangian submanifold>,
$$
\boxed{\operatorname{graph}(\beta)\text{ is Lagrangian}\iff d\beta=0.}
$$
This is the <graph of a closed one-form is Lagrangian> criterion. The opposite conventional sign for the <symplectic form> gives the same criterion.
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