  We introduce a class of integral theorems based on cyclic functions and
Riemann sums approximating integrals. The Fourier integral theorem, derived as
a combination of a transform and inverse transform, arises as a special case.
The integral theorems provide natural estimators of density functions via Monte
Carlo methods. Assessments of the quality of the density estimators can be used
to obtain optimal cyclic functions, alternatives to the sin function, which
minimize square integrals. Our proof techniques rely on a variational approach
in ordinary differential equations and the Cauchy residue theorem in complex
analysis.
