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The work is a single integral over $|r_1-r_2|$, not a double integral over $r_1$ and $r_2$. As you are fixing particle 1, you shouldn't integrate over particle 1. Moreover, the work is $w(r) = \int Fdr$, not $w(r) = \int Frdr$, as you can see from dimensional analysis ($dr$ has the dimension of length, too). Therefore, you treat $r_1$ as constant, integrate ...


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