Hi, I need to integrate a function on an n-dimensional sphere surface. One way is to use the triangle function like: http://en.wikipedia.org/wiki/N-sphere#Spherical_volume_element, however, it is too complex, do you have any idea how to solve it?

My problem is $\int_{\|{\bf U}\| = 1} \sqrt{ {\bf U}^T {\bf M} {\bf U} } d u_1 \ldots d u_n$, where ${\bf M}$ is a positive semidefinite matrix.