Consider the integral $$ \begin{equation} u(x):= \lim_{\eta \to 0^+}\int_{-\infty}^\infty \frac{d s}{s^2 + 1} \left( \frac{1}{x - i \eta - \sqrt{s^2 + 1}} - \frac{1}{x - i \eta + \sqrt{s^2 + 1}} \right) . \end{equation} $$ for $x \in \mathbb{R}$.
Mathematica will evaluate this integral without complaint
In[] : = Assuming[x \[Element] Reals && \[Eta] > 0, Limit[Integrate[1/(s^2 + 1) (1/(x - I \[Eta] - Sqrt[s^2 + 1]) - 1/(x - I \[Eta] + Sqrt[s^2 + 1])), {s, -\[Infinity], \[Infinity]}], \[Eta] -> 0, Direction -> "FromAbove"]]
and returns
$$
u_\mathrm{MMA}(x) = -\frac{4 \arccos\left(\sqrt{1-x^2}\right)}{\left| x\right| \sqrt{1-x^2} }
$$
However this answer cannot be correct, it is an even function ($u(-x) = u(x)$), whereas the definition manifestly has the propery $u(-x) = u(x)^*$ (this can be confirmed using NIntegrate
for small finite values of $\eta$). I.e. both the real and imaginary parts of $u_\mathrm{MMA}(x)$ are even, when the imaginary part should be odd.
Is there a way of getting Mathematica to return the correct answer for these types of integral?