I am trying to show that,
\begin{align} \label{eq:q1main} g(u) = \int_0^1 (1-r^2) J_0(ur)\,\mathrm{d}r = \frac{2}{u^2}J_2(u) \end{align}
Bessel's function of first kind with zero degree is:
\begin{align*} J_0(ur) = \frac{1}{\pi} \int_0^\pi \cos(ur\sin\phi)\,\mathrm{d}\phi \end{align*}
Substitute and evaluate first wrt $r$.
\begin{align*} g(u) &= \frac{1}{\pi} \int_0^1 (1-r^2) \int_0^\pi \cos(ur\sin\phi)\,\mathrm{d}\phi \,\mathrm{d}r \\ &= \frac{2}{\pi} \int_0^\pi \frac{\sin(u\sin\phi)+u\sin\phi\cos(u\sin\phi)}{u^3\sin^3\phi} \,\mathrm{d}\phi \\ \end{align*}
I know that,
\begin{align*} J_2(u) = \frac{1}{\pi} \int_0^\pi \cos(2\phi-u\sin\phi)\,\mathrm{d}\phi \end{align*}
but I cannot show the relation.