# Integral of Dirac function from 0 to a... value

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## Main Question or Discussion Point

Hi. So I'm trying to use Laplace transforms in inverting a particular s-function via the convolution formula.

I ended up with this terrifying-looking thing:

So distributing, I ended up with:

Evaluating the second integral poses no problem for me (although I think the integration will definitely be 'hairy'). I have a problem with the first integral though. How on earth do I integrate the dirac delta? Help! I am totally at a loss here. Or am I doing something wrong?

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mathman
What is your problem? The integral should be the value of the integrand (without the delta function) at t' = 0.

What is your problem? The integral should be the value of the integrand (without the delta function) at t' = 0.
sin(wt) right?

But I did partial fraction expansion using algebra software (mathcad) and the answer was different numerically (I mean, I numerically integrated that function, taking into account that the first integral is sin(w*t)

mfb
Mentor
You might want half that value if your integration border is right where the delta is.
It depends on how the delta appears in the Laplace transform.

BvU