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Let be an interal on $ \mathbb{R}^{n} $ in the form

$$ \int_{R^{n}}dV \frac{P(x_1,x_2,\dots,x_n)}{Q(x_1,x_2,\dots,x_n)}$$

which is divergent.

My question is: Could we substract from this integral a polynomial of the form

$$ K(x_1,x_2,..,x_n) $$ such that the integral

$$ \int_{R^{n}} \left(\frac{P(x_1,x_2,\dots,x_n)}{Q(x_1,x_2,\dots,x_n)}-K(x_1,x_2,x_3,\dots,x_n)\right)dV$$

is now finite?

For the case of a single variable I know how to do it but for multivariable calculus I do not know if this can be done, thanks.

This polynomial $$K(x_1,x_2,x_3,\dots,x_n)$$ can also include inverse power terms for example $ (x_2+1)^{-1}x_3 $ or $ ((x_1+2)(x_2+3))^{-1} $, Laurent polynomials.

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