Compute $$ \int_{-\infty}^{\infty} t^2 \delta(\sin(t)) e^{-|t|} \mathrm dt $$ In closed form, where $\delta(t)$ is the Dirac Delta function .
My attempt:
$$ \int_{-\infty}^{\infty} t^2 \delta(\sin(t)) e^{-|t|} \mathrm dt = \int_{-\infty}^{\infty} \delta(\sin(t))t^2 e^{-|t|}\mathrm dt $$
Then noting that $\sin(t)$ is zero whenever $t=n\pi$. By formula (2) and (7) in the above link,
\begin{align} \int_{-\infty}^{\infty} \delta(\sin(t))t^2 e^{-|t|} \mathrm dt& = \sum_{n=-\infty}^{\infty} \frac{(n\pi)^2e^{-|n\pi|}}{|\cos(n\pi)|} \\& =2\pi^2\sum_{n=0}^{\infty} \frac{(n)^2e^{-n\pi}}{1} \end{align}
However , I am stuck here, i do not know how to procede, Wolfram Alpha tells me that this sum doesn't converge so how can i compute it in closed form? I can only assume I have gone about this the wrong way or made a mistake. Any help would be great.