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T. Clausen, Über die Function sinφ+(1/22)sin2φ+(1/32)sin3φ+etc., J. Reine Angewandte Mathematik 8, 298-300 (1832).

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Article

Evaluation of Some Non-trivial Integrals from Finite Products and Sums

1Institute of Physics, University of Brasilia, P.O. Box 04455, 70919-970, Brasilia DF, Brazil


Turkish Journal of Analysis and Number Theory. 2016, Vol. 4 No. 6, 172-176
DOI: 10.12691/tjant-4-6-5
Copyright © 2016 Science and Education Publishing

Cite this paper:
F. M. S. Lima. Evaluation of Some Non-trivial Integrals from Finite Products and Sums. Turkish Journal of Analysis and Number Theory. 2016; 4(6):172-176. doi: 10.12691/tjant-4-6-5.

Correspondence to: F.  M. S. Lima, Institute of Physics, University of Brasilia, P.O. Box 04455, 70919-970, Brasilia DF, Brazil. Email: fabio@fis.unb.br

Abstract

In this note, by manipulating the sums obtained from certain finite products of trigonometric functions at rational multiples of π, 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.

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