Let $V$ be a smooth one-dimensional confining potential, and suppose that for an energy $E$ the classically allowed region is a single interval $[a(E),b(E)]$ with simple turning points. In the semiclassical regime $\hbar\to0^+$ with the action scale fixed, bound-state energies $E_n(\hbar)$ satisfy the leading asymptotic eigenvalue condition \begin{align*} \int_{a(E)}^{b(E)} \sqrt{2m(E-V(x))}\,dx = \pi\hbar\left(n+\frac{1}{2}\right), \end{align*} up to lower-order corrections in $\hbar$, with $n=n(\hbar)$ varying so that the action remains $O(1)$.