= Lorentzian peak approaching an integration endpoint
{c}
{title2=$\int_0^1[(\lambda x+1-\lambda)^2+\epsilon^2c(x)^2]^{-1}\,dx$}
When $c(0)>0$ and $\lambda-1=O(\epsilon)$, the change $x=\epsilon X$ resolves a truncated <Cauchy distribution>. Its leading contribution is $[\epsilon\lambda c(0)]^{-1}[\pi/2+\arctan((\lambda-1)/(\epsilon c(0)))]$. It matches the nonresonant endpoint integral on one side and the full interior peak on the other. An interior peak at $x_*$ has width $\epsilon c(x_*)/\lambda$; proximity to an endpoint must be measured against this width, not against $\epsilon$ alone.
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