Dirac delta of x Just so there are no misunderstandings let me first ask whether it is true that:
$$
\int_{-\infty}^{\infty}x\delta(x)\mathrm{d}x=0.
$$
If that is not true, then I don't know anything about the Dirac delta distribution and I will be off to correct this :)
Otherwise I have this question: Why can we take a delta functional of $x$? As far as I've read, the Dirac delta is a tempered distribution and it must act on Schwartz functions. I am not convinced that this is true for $f(x)=x$.

You're right, it is not strictly correct to apply - in general - a distribution with $x \mapsto x$ because it is not a compactly supported function.
But the Dirac is itself a compactly supported distribution, so you can extend its domain to all $\mathcal{C}^\infty$ functions.
