# Integrating with a dirac delta function

1. Dec 23, 2012

### fred_91

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

I have to integrate:

$\int_0^x \delta(x-y)f(y)dy$

2. Relevant equations

3. The attempt at a solution

I know that the dirac delta function is zero everywhere except at 0 it is equal to infinity:
$\delta(0)=\infty$
I have to express the integral in terms of function f only and i am unsure how to do this.
Do I have to use integration by parts?

Thank you very much in advance

2. Dec 23, 2012

### Mandelbroth

Dirac's Delta is...weird. You have to integrate $\int_0^x \delta(x-y)f(y)dy$?

Integration by parts sounds...okay? However, one of the properties of the Dirac Delta "function" is that $\displaystyle \forall ε > 0, \int_{a-ε}^{a+ε} f(y)\delta(y-a) \ dy = f(a)$. Can you manipulate your integral into that form?

By my calculations, you should end up with $f(x)(2H(x)-1)$, where H is the antiderivative of δ.

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