= Separable integrand
{title2=$\int_{\prod_j I_j}\prod_jg_j(x_j)\,dx=\prod_j\int_{I_j}g_j(t)\,dt$}
An integrand is separable across coordinates when it factors as $g(x_1,\ldots,x_d)=\prod_{j=1}^dg_j(x_j)$. 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,
$$
\int_{[0,1]^d}\sum_{i=1}^m e^{\|x-a_i\|^2}\,dx
=\sum_{i=1}^m\prod_{j=1}^d\int_0^1e^{(t-a_{ij})^2}\,dt.
$$
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.
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