This question is inspired from an answer given to this question in the physics stackexchange, specifically the integration step going from (12) to (13). We have a distribution given as
$$-f'(x-a)=\sum_{n=0}\frac{C_n}{n!}\delta^{(n)}(x-a)$$
where $\delta$ is the Dirac delta. Suppose that I want to integrate then
$$-\int_x^\infty dx' f'(x'-a)=\int_x^\infty dx' \sum_{n=0}\frac{C_n}{n!}\delta^{(n)}(x-a)$$
where if $C_0=1$ and $f(\infty)=0$ we get
$$f(x-a)=\theta(a-x)-\sum_{n=1}\frac{C_n}{n!}\delta^{(n-1)}(x-a)$$
My questions are:
Where does the minus sign come from? I understand that $\int dx \delta^{(n)}f(x)=(-1)^nf^{(n)}(0)$, but this seems a little different.
Is this kind of treatment legitimate, or is it more of an abuse of the delta function?