Integral of unit impulse function?

1. Apr 30, 2017

Abdulwahab Hajar

1. The problem statement, all variables and given/known data

let's use this symbol to denote the unit impulse function δ
When integrating the unit impulse function (from negative infinity to infinity) ∫δ(t) dt I know that this results in a value of 1 and is only nonzero at the point t = 0.

However for example take this integral into consideration ∫δ(t) e-jωt
since the delta function is only nonzero at the point zero, we only evaluate this multiplication at the point 0 which yields e0 which is 1.

but how can we do that, the integral involves two functions dependant on time shouldn't we integrate from limits for example 0- to 0+ and integrate it by parts or something like that?

2. Relevant equations

∫δ(t) = 1 at t =0
3. The attempt at a solution

My attempt is attempting to explain it above
Thank you

2. Apr 30, 2017

FactChecker

There are a few different ways to put the Derac delta function (generalized function, distribution) on a solid theoretical basis (see https://en.wikipedia.org/wiki/Dirac_delta_function). The result and goal of all of them is that ∫δ(t)f(t)dt = f(0).

3. Apr 30, 2017

Abdulwahab Hajar

thank you, you were very helpful....
loving the profile pic btw!!!