Changing signs of integration limits

In fact, there is another sign change involved. Formally, it is best to perform the substitution $u = -x$. Then $du = -dx$ and so

$$ \int_{-R}^{-\varepsilon} \frac{e^{ix}}{x} \, dx = \int_{R}^{\varepsilon} \frac{e^{-iu}}{-u} \, (-du) = \int_{R}^{\varepsilon} \frac{e^{-iu}}{u} \, du = -\int_{\varepsilon}^{R} \frac{e^{-iu}}{u} \, du.$$

You will use

$$* \int_a^b f = -\int_b^af $$

and the substitution formula

$$** \int_a^bf (\phi (t))\phi'(t)dt=\int_{\phi (a)}^{\phi (b)}f (x)dx $$

with $\phi (t)=-t $

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