= Quadratically weighted inverse-radius Hessian integral
{title2=$\int_{r<a}x_ix_j\partial_k\partial_l(r^{-1})\,dV=\frac{2\pi a^2}{15}(-2\delta_{ij}\delta_{kl}+3\delta_{ik}\delta_{jl}+3\delta_{il}\delta_{jk})$}
Away from the origin the <Hessian matrix> of $r^{-1}$ is $(3x_kx_l-r^2\delta_{kl})/r^5$. Multiplication by $x_ix_j$ makes the singularity locally integrable. Rotational invariance and the symmetries in each index pair reduce the integral to $A\delta_{ij}\delta_{kl}+B(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$. The Laplacian trace gives $3A+2B=0$, while contracting the two pairs as $k=i,l=j$ gives $3A+12B=\int_{r<a}2/r\,dV=4\pi a^2$. Solving these two equations yields the displayed expression.
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