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are formal manipulations of divergent integrals admisible whenever the integral are divergent [tex] \infty [/tex]

i mean if i have a 4-dimensional integral [tex] \int_{R^{4}}dxdydzdt F(x,y,z,t) [/tex]

why can we make a change of variable to polar coordinates ?? , or for example if we have an UV divergent integral [tex] \int_{0}^{\infty}dxx^{4} [/tex] by means of a change of variable [tex] x= 1/y [/tex] this integral is IR divergent [tex] \int_{0}^{\infty}dyy^{-6} [/tex] or if i have the divergent integral

[tex] \int_{0}^{\infty}\int_{0}^{\infty}dxdy \frac{(xy)^{2}}{x^{2}+y^{2}+1} [/tex]

this is an overlapping divergence but if i change to polar coordinates then i should only care about [tex] \int_{0}^{\infty}dr \frac{r^{3}}{r^{2}+1} [/tex] which is just a one dimensional integral.

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# Are mathematical manipulations admisible if integral are divergents ?

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