{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:57:00Z","timestamp":1760144220550,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,3,31]],"date-time":"2024-03-31T00:00:00Z","timestamp":1711843200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Zhoukou Normal University high-level talents start-up funds research project","award":["ZKNUC2022007"],"award-info":[{"award-number":["ZKNUC2022007"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Making use of integration by parts and variable replacement methods, we derive some interesting explicit definite integral formulae involving trigonometric or hyperbolic functions, whose results are expressed in terms of Catalan\u2019s constant, Dirichlet\u2019s beta function, and Riemann\u2019s zeta function, as well as \u03c0 in the denominator.<\/jats:p>","DOI":"10.3390\/axioms13040230","type":"journal-article","created":{"date-parts":[[2024,3,31]],"date-time":"2024-03-31T13:32:56Z","timestamp":1711891976000},"page":"230","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Exploring Explicit Definite Integral Formulae with Trigonometric and Hyperbolic Functions"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2287-5196","authenticated-orcid":false,"given":"Yulei","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9226-254X","authenticated-orcid":false,"given":"Dongwei","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1007\/s00208-003-0420-2","article-title":"Diophantine properties of numbers related to Catalan\u2019s constant","volume":"326","author":"Rivoal","year":"2003","journal-title":"Math. Ann."},{"key":"ref_2","first-page":"1","article-title":"M\u00e9moire sur la transformation des s\u00e9ries et sur quelques int\u00e9grales d\u00e9finies","volume":"33","author":"Catalan","year":"1865","journal-title":"M\u00e9m. Acad. R. Sci. Belg."},{"key":"ref_3","unstructured":"Bradley, D.M. (2023, July 08). Representations of Catalan\u2019s Constant. Available online: https:\/\/www.researchgate.net\/publication\/2325473."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1080\/10652460701688125","article-title":"Closed-form evaluation of some families of cotangent and cosecant integrals","volume":"19","year":"2008","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1083","DOI":"10.1007\/s10114-020-9451-9","article-title":"Identities for Catalan\u2019s constant arising from integrals depending on a parameter","volume":"36","author":"Ferretti","year":"2020","journal-title":"Acta Math. Sin."},{"key":"ref_6","first-page":"385","article-title":"Problem 12317","volume":"129","author":"Stewart","year":"2022","journal-title":"Am. Math. Mon."},{"key":"ref_7","first-page":"587","article-title":"Problem 12332","volume":"129","author":"Holland","year":"2022","journal-title":"Am. Math. Mon."},{"key":"ref_8","unstructured":"Prudnikov, A.P., Brychkov, Y.A., and Marichev, O.I. (1986). Integrals and Series, Volume 1. Elementary Functions, Gordon & Breach Science Publishers."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Andrews, G.E., Askey, R., and Roy, R. (1999). Special Functions, Cambridge University Press.","DOI":"10.1017\/CBO9781107325937"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/4\/230\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:21:45Z","timestamp":1760106105000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/4\/230"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,31]]},"references-count":9,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["axioms13040230"],"URL":"https:\/\/doi.org\/10.3390\/axioms13040230","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,3,31]]}}}