Summary: The integral identity found by {\it M. L. Glasser\/} [J. Phys. A, Math. Theor. 44, No. 22, Article ID 225202, 5 p. (2011; Zbl 1252.33012)] is generalized using the permutation symmetry of coordinates of an $n$-spherical surface simplex. The first calculation technique is simple to apply, but the second technique allows further generalization of M. L. Glasser’s identity. Analogous results are discussed for the $n$-hemispherical surface of the unit $n$-sphere and for the entire surface of the $n$-sphere. The $n$-sphere surface result is used to generalize M. L. Glasser’s solution to a problem proposed by {\it J. R. Bottiger\/} and {\it D. K. Cohoon\/} [“A normalization constant", SIAM Review 29, 302‒303 (1987)].