I want to find out if this integral can be calculated (if it exists)
$$\int_{[0,1]^3}^{} (x^3+y^2)z^{-1}d(x,y,z)$$
To be honest, I don't know how, but I think that one might has to use Fubini's theorem since this is an iterated integral. Does someone know how it's done?
$$I=\int_{[0,1]^3}\frac{x^3+y^2}z\mathrm{d}(x,y,z)$$ $$I=\int_{[0,1]}\int_{[0,1]}\int_{[0,1]}\frac{x^3+y^2}z\mathrm{d}x\mathrm{d}y\mathrm{d}z$$ $$I=\int_{[0,1]}\int_{[0,1]}\int_{[0,1]}\frac{x^3}z\mathrm{d}x\mathrm{d}y\mathrm{d}z+\int_{[0,1]}\int_{[0,1]}\int_{[0,1]}\frac{y^2}z\mathrm{d}x\mathrm{d}y\mathrm{d}z$$ $$I=\int_{[0,1]}x^3\mathrm{d}x\int_{[0,1]}\mathrm{d}y\int_{[0,1]}\frac{\mathrm{d}z}z+\int_{[0,1]}\mathrm{d}x\int_{[0,1]}y^2\mathrm{d}y\int_{[0,1]}\frac{\mathrm{d}z}z$$ $$I=\frac14\cdot1\cdot\big(\log1-\log0\big)+1\cdot\frac13\cdot\big(\log1-\log0\big)$$ $I$ does not converge :(