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Can we meaningfully ascribe values to these divergent integrals?

Depends on how you choose to define it :) A natural way is to define $\int_0^{\infty} = \lim_{a\rightarrow0} \int_a^{1/a}$, in which case: \begin{eqnarray*} \int_0^{\infty}\left(1-\frac1{x^2}\right)dx …
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