Integrating with a dirac delta function

[fred_91]

Homework Statement

I have to integrate:

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

Homework Equations

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

 

[Mandelbroth]

Homework Statement

I have to integrate:

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

Homework Equations

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

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.