Integrating ∫x·eˣ dx by parts works; ∫eˣ·x dx with the choices reversed produces a harder integral. The integrand is identical. What does that show about the technique?

C. That it is a tool requiring judgement rather than a procedure

Both applications are valid; only one makes progress, because differentiating x simplifies it and differentiating eˣ does not. LIATE is a heuristic encoding that observation, and it is why integration is a craft where differentiation is an algorithm.