 Separable integrand

An integrand is separable across coordinates when it factors as $g(x_1,\dots ,x_d)={\prod }_{j=1}^d{g}_j(x_j)$. On a product domain, the Fubini theorem reduces its integral to a product of one-dimensional integrals, provided the factors are integrable; for nonnegative factors the Tonelli theorem gives the extended-value version.

A finite sum of separable terms can likewise be integrated as a finite sum of products. For example,For fixed centres, the factors are smooth on compact intervals. Repeated one-dimensional composite Simpson's rule refinements have fourth-order discretization error, avoiding the exponential tensor-grid cost of a generic $d$-dimensional rule. The numerical constants and possible floating-point overflow still depend on the scale of the factors; separability removes a dimension-dependent combinatorial cost, not all conditioning difficulties.