can you integrate the dirac delta over any domain? At eq 28 the following author takes it from 1 to infinity. I've never seen this before and i'm not sure his equation is correct.
http://bado-shanai.net/The%20Table%20of%20Integrals/intResiduesofDiracDelta.htm
 A: An integral of the form
\begin{equation}
\int_1^∞\delta(x)f(x)dx
\end{equation}
is just $0$. Intuitively, the spike of the Dirac delta is not in the domain, so the integral only picks up $0$.
If you look closer at the integral you mention:
\begin{equation}
\int_1^∞x^{-4}\delta[\sin (\pi x)]\cos(\pi x)\pi dx 
\end{equation}
you will see that the argument in the delta is in fact oscillating between $-1$ and $1$. The value of the integral is just the sum of $x^{-4}\cos(\pi x)\pi$ divided by $\pi$ whenever $\sin (\pi x)=0$ that is for $x=1,2,3...$
\begin{equation}
\sum_{x=1}^∞\frac{1}{x^4}\cos(\pi x)=\sum_{x=1}^∞\frac{(-1)^x}{x^4}
\end{equation}
which is in fact different from what your source states. In fact I would have:
\begin{equation}
\int_1^∞x^{-4}\delta[\sin (\pi x)] \pi dx = \sum_{x=1}^∞\frac{1}{x^4}
\end{equation}
but the author justifies the cosine by saying it "comes from the derivative in the functional differential". Perhaps you know what it means, or somebody can correct my mistakes.
