Here is the link:https://en.wikipedia.org/wiki/Fundamental_solution#Application_to_the_example
It states that :
$$∫_{-∞}^∞\frac{1}{2} |x-y| \sin(y) dy=-sin(x)$$ as distributions
The best I can come up with is to prove that:
⟨$∫_{-∞}^∞$$1/2 |x-y|$ sin$y$$dy$, $ϕ$⟩=$⟨-sin(x),ϕ⟩$$∀ϕ$ test function
But as far as I can tell the convolution: $∫_{-∞}^∞$$1/2 |x-y|$ sin$y$$dy$ is not possible even in the sense of distributions.