perhaps it will make more sense if you realize that .
this is a consequence of the chain rule: is really the composition:
, where g(x)= -x = (-1)x.
so when we integrate, we need a minus sign to go inside the integral, and another minus sign to go outside (to balance, (-1)(-1) = 1). the minus sign inside the integral disappears, since when we integrate:
.
the minus sign outside the integral stays, however.