This note (PDF version) discusses a generalization of some formulas in "A Heuristic for Exact Calculation of n-Dimensional (n+1)-Parametric Integrals of Some Elementary and Special Functions" by Vladimir V. Bondarenko.  The following simple theorem can be used in combination with the more interesting results appearing in the next two sections.

Theorem. Suppose and .
Then .

An n-Dimensional F Integral

Theorem. Assume is not a negative integer and is a positive integer.  Then

Comment. Since

for are generated by a finite dimensional basis over , the indefinite integrals

satisfy a linear recurrence relation.  This linear recurrence relation can be explicitly computed.

Example.

Example.

An n-Dimensional G Integral

Theorem. Assume is a positive integer. Then

Definite Integrals

Definition. Let

Theorem. Assume .  Then