Two-term Dirichlet WKB quantization (${ϵ}^{-1}{\int }_a^{b}Kdx+ϵ{\int }_a^b{\sigma }_{1}dx=n\pi +O({ϵ}^3)$)

For smooth positive $K$ on an interval with regular Dirichlet boundary conditions, there is no turning-point phase shift. Combining conjugate WKB approximation branches to vanish at the first endpoint makes the total phase at the second an integer multiple of $\pi$. The first correction is the integrated WKB phase correction for a slowly varying oscillator. For $K=x+1$ on $[0,1]$ the resulting eigenvalues are $4{\pi }^2{n}^2/9-3/16+O(n^{-2})$.