I need to calculate the following integral
$$ \int_0^{\infty}xdxJ_n(kx) $$
Integrating it by parts and using the normalization of Bessel functions, I find it (somewhat heuristically) to equal the Dirac Delta function $\delta(k)$. I cannot find a single source online or in a reference like Abramowitz and Stegun to back this up. Can someone here confirm this?
The integral $$ \int_{0}^{+\infty} x\, J_n(x)\,dx $$ is not converging for any $n\in\mathbb{N}$, since $J_n(x)$ decays like $\frac{1}{\sqrt{x}}$ as $x\to +\infty$.
On the other hand, for any $n\in\mathbb{N}$ we have: $$ \int_{0}^{+\infty} J_n(x)\,dx = 1 $$ since: $$ \mathcal{L}(J_n(x)) = \frac{1}{\sqrt{1+s^2}\left(s+\sqrt{1+s^2}\right)^n}.$$