I am trying to integrate the following problem with limit +D and -D

Now intuition tells me to treat it like

Here is where I'm going wrong - the way I'm integrating gives me

But if I differentiate that I get

So have randomly created a factor of

. Therefore I'm doing something wrong but I can't seem to get a rule to fit the situation.

Differentiating

Returns me to the original problem

But that would imply that the basic integral is divided by

despite the fact that for all intents and purposes

is a constant.

Can anyone explain, as I think I've horribly confused myself about something not that complex?