Giving meaning to an invalid integral calculation
I know that the following calculation is invalid because the integrand is
not continuous over the interval $-1 \leq x \leq 1$, but apparently the
final result can still be given meaning. $$\int_{-1}^{1}\frac{dx}{x^{2}} =
\left [ -\frac{1}{x} \right ]_{-1}^{1} = -1 - 1 = -2$$
Supposedly a clue lies in the fact that $$\int_{-\infty
}^{-1}\frac{dx}{x^{2}} = \int_{1}^{\infty }\frac{dx}{x^{2}} = 1$$
but I still have no idea what meaning can be gleaned from the result.
Help me out? Thanks (:
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