Science and Education Publishing Turkish Journal of Analysis and Number Theory 2333-1232 4 6 2016 11 26 Evaluation of Some Non-trivial Integrals from Finite Products and Sums 172 176 EN F. M. S. Lima Institute of Physics, University of Brasilia, P.O. Box 04455, 70919-970, Brasilia DF, Brazil TJANT2016465 10.12691/tjant-4-6-5 2016 8 3 2016 9 20 2016 11 24 In this note, by manipulating the sums obtained from certain finite products of trigonometric functions at rational multiples of , I put them in the form of Riemann sums. By taking the limit as the number of (equally-spaced) subintervals tends to infinity, I have found exact closed-form results for some non-trivial integrals, e.g. and I also show how the method applies for the prompt evaluation of more complex integrals, such as and Since this approach does not involve any search for primitives, it can be a good alternative to more complex integration techniques.