It is all the same. Please ponder it until you see that it is not scary.
Feel free simply to remove constants from the integral argument. is a constant.
By the way, "the derivative of e" is zero. Be more careful with words.
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.