It’s interesting he mentions he doesn’t like contour integration since many integrals can be done either way.

Feynman’s trick is equivalent to extending it into a double integral and then switching the order of integration.

Don't forget to check for the necessary measurability & integrability of the sections (f(a, y), f(x, b)) before switching the order: https://en.wikipedia.org/wiki/Fubini%27s_theorem?useskin=vec....