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Relations between integrals arising from Baillie’s identity on trigonometric series
Published online by Cambridge University Press: 15 October 2025
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Baillie’s identity is 
(see [1] or [2], p. 240). Its integral analogue, 
, is not difficult to prove (see Lemma 1, below). In this Article, we prove a generalisation of the latter result (see Theorem 1). Theorems 2 and 3 are extensions, involving in addition, powers of cosines in the integrand. Theorem 4 answers a question raised after the proof of Theorem 1, and Theorem 5 collects together the preceding results in the form of three identities between trigonometric integrals. Theorem 6 gives a further generalisation.
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- © The Authors, 2025 Published by Cambridge University Press on behalf of The Mathematical Association
References
Baillie, Robert, Solution to Advanced Problem 6241 (also proposed by R. Baillie), Amer. Math Monthly 87 (1980) pp. 496–497.10.2307/2320278CrossRefGoogle Scholar
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