Where do you get that from?

The definite integral int -oo^+oo f(t).d(t-T) gives us f(T), but there is no information given to us about the indefinite integral nor of the anti-derivative.

Indeed, the definite integral gives us f(T), which is a constant function having no aspect of t in its evaluation and which graphs as a horizontal line from -oo to +oo. This comes from the basic definitions of the Diracian.

Your claimed anti-derivative has a strong dependence on t, and therefore has not come from anything formally defined.

Are you making this up as you go along?

Well, here is what David C. Ullrich posted..