I'm trying to understand a derivation (Risken, "The fokker-planck equation", 2nd ed. page 66). Everything was going smoothly until
The negative sign of the differential $\mathrm{d}\Delta = - \mathrm{d}x'$ may be absorbed into the integration boundaries.
In the case at hand, the integration boundaries are supposed to be $-\infty$ and $+\infty$. I've always been taught since high school that the integral boundaries define an ordering, that is $\int_b^a=-\int_a^b\neq\int_a^b$, no matter what the underlying measures are (Riemann, Lebesgue, Radon, ...) and whether $a$ and $b$ are finite or not. The "$-$" sign remains. So what does it mean that they absorb a negative sign?
I report below the relative page(s) of the book
The point you appear to be missing is that the change of variable also flips the bounds of integration. Consider an integral $$\int_{-\infty}^\infty f(t)\,dt.$$ If you make the substitution $t=-u$, this replaces $dt$ with $-du$, but it also replaces $\int_{-\infty}^\infty$ with $\int_\infty^{-\infty}$, since as $t\to\pm\infty$, $u\to\mp\infty$. So the integral becomes $$\int_{\infty}^{-\infty}-f(-u)du.$$ But now we can flip back the bounds of integration by adding another minus sign, so the end result is $$\int_{-\infty}^\infty f(t)\,dt=\int_{-\infty}^\infty f(-u)\,du.$$ Here at first glance it looks like you forgot to replace $dt$ with $-du$, but actually you did and this cancelled out with the effect the substitution had on the bounds of integration.