Integration of rational functions over $\mathbb R^n$ by means of toric compactifications and multidimensional residues

Two methods for computing the integrals of rational functions over $\mathbb R^n$ are considered. The first is applicable to differentials with rational antiderivatives and uses the interpretation of $\mathbb R^n$ as a chain of integration in some toric compactification. The second method is based on the theory of multidimensional residues and the multidimensional version of the Sokhotskii formula for the jump of an integral.