The integral of a function $f$ defined on a symmetric space $M \simeq G/K$ may be expressed in the form of a determinant (or Pfaffian), when $f$ is $K$-invariant and, in a certain sense, a tensor power of a positive function of a single variable. The paper presents a few examples of this idea and discusses future extensions. Specifically, the examples involve symmetric cones, Grassmann manifolds, and classical domains.