I need Help understanding how to integrate the following functions with the dirac deltas.
$$ \int_{-1}^1 (\delta(x)\cos(x) + \delta(x-2) x^2) dx $$
I Feel like im over complicating this but i have no clue what to do with this. Would I use the property that says dirac delta(x)*f(x) = dirac delta(x)*f(0)? Please help, Thank You!
First of all, you can split this into two integrals, because integrals are linear. The second integral will be zero, because all the mass of the dirac delta function is situated in the place where $x = 2$, but that is not in the domain. For the other integral, you can indeed use the theorem as mentioned. So, $$ \int_{-1}^1 (\delta(x)\cos(x) + \delta(x-2) x^2) dx = \int_{-1}^1 \delta(x)\cos(x) dx + \int_{-1}^1 \delta(x-2) x^2 dx = \int_{-1}^1 \delta(x)\cos(x) dx = cos(0) = 1. $$
So, the most important thing is that within $[-1,1]$ it cannot happen that $x-2$ becomes $0$, so the second integral is $0$.