Feynman's path-integral - derivation of the average of energy estimator over probability distribution

Question

This is directly from Tuckerman's Statistical Mechanics textbook, Chapter 12.6

The part that gets me confused starts from the 2nd line of eqn. (12.6.38) to the first line of eqn.(12.6.39), I can't follow how it's derived? It would be much appreciated if I can get some hints here!

Answer 1

As physicists like doing, that's integration by parts, and neglecting the boundary term.