= Logarithmic spherical Bessel square integral
{title2=$\int_0^\infty j_\ell(x)^2\,dx/x=1/[2\ell(\ell+1)]$}
For integer $\ell\geq1$, put $I_\ell=\int_0^\infty j_\ell(x)^2\,dx/x$ and $J_\ell=\int_0^\infty j_\ell(x)j_{\ell-1}(x)\,dx$. Integrating $j_\ell'=j_{\ell-1}-(\ell+1)j_\ell/x$ gives $J_\ell=(\ell+1)I_\ell$. For $\ell>1$, integrating $j_{\ell-1}'=(\ell-1)j_{\ell-1}/x-j_\ell$ also gives $J_\ell=(\ell-1)I_{\ell-1}$. For $\ell=1$, the boundary values $j_0(0)=1$ and $j_0(\infty)=0$ give $J_1=1/2$, hence $I_1=1/4$. The recurrence proves the displayed identity. The integral without the square is different; for example $\int_0^\infty j_1(x)\,dx/x=\pi/4$.
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