Let $f(x)$ be an any real-valued function, and let $x_0 \ne 0$. I want to evaluate the following two integrals:
1) $\int_{-\infty}^{\infty}f(x)\delta(x-x_0)\delta(x+x_0)dx$
2) $\int_{-\infty}^{\infty}f(x)\delta(x-x_0)\delta(x-x_0)dx$
For 1), I think that I can simply use the sifting property of the delta function, i.e.
$$\int_{-\infty}^{\infty}f(x)\delta(x+x_0)\delta(x-x_0)dx = f(x_0)\delta(x_0+x_0) = f(x_0)\delta(2x_0)$$
Since $\delta(x) = 0$ if $x \ne 0$, then $\delta(2x_0)=0$, which means that $\int_{-\infty}^{\infty}f(x)\delta(x+x_0)\delta(x-x_0)dx=0$.
For 2), I am tempted to say that $\int_{-\infty}^{\infty}f(x)\delta(x-x_0)\delta(x-x_0)dx = f(x+x_0)$, but I am not sure.
I can't find an explanation or a rigorous proof for either integral. Any help would be appreciated... Thanks!