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How to integrate the $L^2$ function $1/|x|$ numerically
Following Michael Renardy's suggestion, write $g=\phi\psi$, and write
$$
g(x)=g(0)+x\cdot G(x) ,
$$
where $G$ is a smooth function. Then the integral becomes
$$
\int_\Omega fg = g(0)\int_\Omega f + \s …