= Endpoint transition of a regularized interior pole
{title2=$I\sim[\pi/2+\arctan(d/(\eta L_0))]/(a\eta L_0)$}
Suppose $a,b,L_0>0$, $L(x)$ is smooth and positive near zero with $L(0)=L_0$, and $d=O(\eta)$ as $\eta\to0^+$. The leading contribution to
$$
I=\int_0^b\frac{dx}{(ax-d)^2+\eta^2L(x)^2}
$$
is obtained by scaling $x=\eta s$ and replacing $L$ by $L_0$. Integration of the resulting reciprocal quadratic gives the displayed formula. For constant $L$ the exact expression is
$$
I=\frac1{a\eta L_0}\left[\arctan\frac{ab-d}{\eta L_0}+\arctan\frac d{\eta L_0}\right].
$$
This proves the half-peak value at $d=0$ and the transition between an interior peak and a pole outside the interval. If $L$ varies, smoothness and positivity on the contributing neighborhood justify its leading replacement; other zeros or singular contributions elsewhere must be excluded.
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