Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11, 274-288 2025 Publisher: Learning Gate DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate © 2025 by the authors; licensee Learning Gate History: Received: 13 August 2025; Revised: 1 October 2025; Accepted: 6 October 2025; Published: 4 November 2025 * Correspondence: omarin@unfv.edu.pe Calculation and application of the engineering numbers of Bernoulli (B_n) and Euler (E_n) to integral analysis and other applications Marín-Machuca, Olegario1*, Candela-Díaz, José Eduardo2, Rojas-Rueda, María del Pilar3, Chinchay-Barragán, Carlos Enrique4, Pérez-Ton, Luis Adolfo5, Vargas-Ayala, Jessica Blanca6 1Academic Department of Food Sciences, Faculty of Oceanography, Fisheries, Food Sciences, and Aquaculture. Universidad Nacional Federico Villarreal. Lima 1500 Peru; omarín@unfv.edu.pe (M.M.O.). 2Faculty of Oceanography, Fisheries, Food Sciences, and Aquaculture. Universidad Nacional Federico Villarreal. Lima, Peru. jcandela@unfv.edu.pe (C.D.J.E.). 3School of Human Medicine, Universidad Norbert Wiener, Lima, Peru. maria.rojasr@uwiener.edu.pe (R.R.M.P.). 4Professional School of Food Engineering, Faculty of Fisheries and Food Engineering, Universidad Nacional del Callao. Callao, Peru. cchinchayb@unac.edu.pe (C.B.C.E.). 5Professional School of Food Engineering, Faculty of Fishery and Food Engineering, Universidad Nacional del Callao, Callao, Peru. laperezt@unac.edu.pe (P.T.L.A.). 6Academic Department of Aquaculture, Faculty of Oceanography, Fisheries, Food Sciences and Aquaculture, Universidad Nacional Federico Villarreal, Lima 15074, Peru. jvargas@unfv.edu.pe (V.A.J.B.). Abstract: The calculation of Bernoulli and Euler numbers is based on the fact that these numbers appear in various mathematical expressions and engineering phenomena. Although computers and software programs can generate large quantities of these numbers, only a limited number are published in current references. Consequently, their use often results in unsatisfactory approximations. The objective was to calculate and evaluate the first twenty (20) Bernoulli and Euler numbers, subsequently applying them to mathematics and proposing their use in environmental engineering. There is a wide variety of relationships involving Bernoulli and Euler numbers that appear in several practical mathematical applications. In both cases, the most effective relationships used in this study are binomial expansions. By employing the appropriate and proposed equations, the Bernoulli and Euler numbers were calculated both manually and mechanically, successfully determining the number of values set as the goal. Finally, Bernoulli and Euler numbers significantly contribute to the estimation of various processes and mathematical applications, such as calculating the integrals of ln⁡(𝑐𝑜𝑠𝑥) and sec⁡(𝑎𝑥) 𝑥⁄ ; respectively. Keywords: Applications, Engineering, Bernoulli-Euler numbers, Integrals. 1. Introduction Benton [1] notes that the Bernoulli numbers originate from a family closely associated with mathematics and engineering for several decades, even offering solutions to problems in mathematical analysis in the field of differential equations and their solution methods. Boyer [2] mentions that Euler numbers appear in the brilliant works of Leonhard Euler on Introduction to Analysis, Institutiones Calculi Differentialis, and Algebra. Courant and John [3] state that the difficulty, or indeed the problem, in determining Bernoulli and Euler numbers lies in the fact that there are scientific processes and phenomena in engineering sciences where functions strongly correlate with these numbers and must be analyzed, evaluated, and interpreted carefully and meticulously. Despite frequent mentions of the extensive range of Bernoulli and Euler numbers, only a limited subset is published, resulting in the use of alternative solutions such as approximation techniques. In this sense, their determination is well justified. Benton [1] reports that https://orcid.org/0000-0002-0515-5875 https://orcid.org/0000-0002-4198-5745 https://orcid.org/0000-0003-3812-7579 https://orcid.org/0000-0003-0053-4865 https://orcid.org/0000-0001-7040-1502 https://orcid.org/0000-0002-8889-0954 275 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate Jacob Bernoulli published his book Ars Conjectandi, where, for the first and only time, a large set of mathematical expressions and relationships was presented, defined as important and suitable formulas for calculating the sums of positive integer powers of the first n natural numbers Abramowitz and Stegun [4] further states that all Bernoulli numbers are positive and, from the fourth onward, grow without bound. This growth can be expressed by the relationship 𝑙í𝑚 𝑚 = ∞ ( 𝐵𝑚 𝐵𝑚−1 ) = 𝑚2 𝜋2 . According to Fernández [5] and Vera [6], the Bernoulli were first used in numerous practical cases by Moivre and Euler. Bronshtein and Semendiaev [7] indicate that Euler numbers are quantitatively considered and generally related through simple and complex relationships, equations, and functions within integral and differential calculus, and in phenomena that interconnect these mathematical concepts. Bronshtein and Semendiaev [7] and Marín [8] mention that both Bernoulli and Euler numbers have practical applications in applied and abstract mathematical sciences across various fields of knowledge and in some engineering science applications. One of the eleven descendants and disciples of the famous Bernoulli family moved to Basel, Switzerland, to continue mathematical training and preserve the legacy of the Bernoulli lineage. Child [9] and Fernández [5] mention that, due to the great importance and versatility of Bernoulli numbers, they were published for the first time in 1713, in the splendid, magnificent, and excellent work titled Ars Conjectandi, after a period of eight years had passed since the death of its great author, Jacob Bernoulli. Ruíz [10] and Martínez et al. [11] indicate, after a rigorous mathematical analysis, that with respect to Bernoulli numbers, the frequent and accurate relationships, in the form of series, are those that contain these everyday, immeasurable, and greatly important numbers. With respect to Bernoulli numbers, the series-form relationships are those that contain these numbers, Martínez et al. [11] and Ruíz [10]. Vega-Calderón et al. [12] indicate that Bernoulli numbers have no direct relationship with Bernoulli's equation, and that this equation, regarding fluid flow, presents certain difficulties for students of subjects that describe fluid flow phenomena or behaviors of water properties. It is also mentioned that Bernoulli numbers were published and initially corroborated as validation and proof in the year 1713, in the work titled Ars Conjectandi, ten years after the death of its progenitor and author, Fernández [5] and Child [9]. Vega-Calderón et al. [12] mention, after extensive and rigorous discussion and analysis, that Bernoulli numbers have no direct or indirect relationship with Bernoulli's equation, widely applied and used in the treatment of phenomena and processes involving the flow of substances with thermophysical properties analogous to water, and the different relationships with said equation, regarding fluid flow, present certain difficulties to be understood and applied by students of subjects dealing with runoff processes and describing fluid flow phenomena, whatever these may be. Sierra Cotes and Gómez Yance [13] indicate that, when studying the behavior of diameter, jet velocity, and flow with different vertical discharge heights, the process followed the trends of Bernoulli number functions, which consisted of the study of flow through an orifice, where the velocity of the fluid is studied analogous to what the Bernoulli numbers and equation describe for the outflow of regulatory boxes. Huancapaza [14] indicates that his objective was to determine which numerical method is most effective for obtaining the dynamic response of multi-degree-of-freedom structures, resulting in Euler numbers describing a Gaussian behavior with increasing time. Torres [15] states in his research, "Generation of a mathematical model for the study of reservoir operation with a multipurpose approach," where the objective was the development of a mathematical model using the principles of normal distribution mechanics to simulate reservoir operations, resulting in Euler numbers and their method being the simplest numerical scheme. Zamora et al. [16] argue that Bernoulli numbers represent asymptotic decay processes regarding the conservation of mechanical energy, allowing analytical and experimental methods to determine different parameters such as tank emptying time, pipe area, flow velocity, and velocity head. Estela Urbina et al. [17] indicate that Bernoulli numbers are very important in asymptotic decay processes, 276 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate being an adequate strategy to solve problematic situations of complex short transitions, considering that friction or drag losses are negligible. Estela Urbina et al. [17] affirm that Bernoulli numbers are very important in asymptotic decay processes, being an adequate strategy to solve problematic situations of complex short transitions, considering that friction or drag losses are negligible. Marín-Machuca et al. [18] mention in their research "Quantification of the alkalinity of natural water applying Bernoulli numbers and Simpson's approximate integral" that the purpose was to calculate the first twenty Bernoulli numbers and apply them to mathematics (∫ ln(𝑐𝑜𝑠𝑥) 𝑑𝑥) and to environmental engineering, quantifying the alkalinity of natural water in the form of calcium carbonate (𝐶𝑎𝐶𝑂3)(𝑚𝑔/𝑙), measuring the carbon dioxide (𝐶𝑂2); that there exists, with certainty and specifically, a great variety of mathematical relationships and expressions that include Bernoulli numbers, appearing in various practical applications of mathematics; that the best relationships used on this occasion are binomial expansions; that by applying the proposed and suitable equations, Bernoulli numbers were satisfactorily calculated, reaching the number of values set as the objective, and finally, Bernoulli numbers decisively and significantly contribute to mathematical processes such as calculating integrals of the type ln⁡(𝑐𝑜𝑠𝑥), as well as applications in environmental engineering, evaluating water alkalinity as calcium carbonate (𝐶𝑎𝐶𝑂3). Child [9] mention that the anecdote told by Gauss (1777–1855) about his early years in primary school is well known, when he impressed his bad-tempered mathematics teacher with a very simple but ingenious procedure for summing the first 100 terms of an arithmetic progression; however, it is not as well known that a hundred years earlier, one of the most outstanding mathematicians of the Bernoulli family, Bernoulli [19] left for posterity an excellent, outstanding, and brilliant work, which could well be considered the first great treatise on combinatorics and probability: Bernoulli [19] where in this brilliant work, highly recommended reading, there appears a result far superior to Gauss's sum, asking: What do you think if we calculate the sum of the first 1000 powers of 10 of the natural numbers? But of course, with pencil and paper, or quill, which were the tools our dear friend, mathematician, and teacher Bernoulli had at that time, kindly presenting in this article some of the essential and fundamental ideas that led him to carry out such purpose and which were also the origin, beginning, and foundation of some numbers that today bear his name: "the Bernoulli numbers." Martínez et al. [11] have developed the main and relevant aspects regarding Bernoulli polynomials and numbers, which serve, due to their importance and foundation, for the presentation of the results found to test, conjecture, and prove the Faulhaber Formula Theorem, in which the cotangent function is developed in a series, and to infinity, in relation to Bernoulli numbers and in this way the values of the Riemann zeta function are evaluated at integer arguments through Bernoulli numbers, and finally the explicit formulas for evaluating and calculating Bernoulli numbers in terms of Stirling numbers of the first and second kind are verified. The purpose has been to determine and evaluate the first twenty Bernoulli and Euler numbers, to present each of them in table form, and to apply them, in some way, in the calculation of integrals with expressions containing said numbers. 𝑥 2 𝑐𝑜𝑡 𝑥 2 = 1 − 𝐵1 𝑥2 2! − 𝐵2 𝑥4 4! − 𝐵3 𝑥6 6! −. .. 𝑥 2 𝑒𝑥+1 𝑒𝑥−1 = 1 + 𝐵1 𝑥2 2! − 𝐵2 𝑥4 4! + 𝐵3 𝑥6 6! −. .. 𝐵𝑚 = (2𝑚)! 22𝑚−1𝜋2𝑚 [1 + 1 22𝑚 + 1 32𝑚 + 1 42𝑚 +. . . ] 𝐵𝑚 = 4𝑚∫ 𝑡2𝑚−1.𝑑𝑡 𝑒2𝜋𝑡−1 𝑡𝑔. 𝑥 = 22(22 − 1)𝐵1 𝑥 2! + 24(24 − 1)𝐵2 𝑥3 4! +. .. 𝑙𝑜𝑔 𝑠𝑒𝑛.𝑥 𝑥 = − 2 1 𝐵1 𝑥2 2! − 23 2 𝐵2 𝑥4 4! − 25 3 𝐵3 𝑥6 6! −. . . .. 277 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate 𝑥 𝑒𝑥−1 = ∑ 𝐵𝑛.𝑥 𝑛 𝑛! ∞ 𝑛=0 = 𝐵0 + 𝐵1𝑥 + 𝐵2 2! 𝑥2 + 𝐵3 3! 𝑥3+. .. 𝑥 𝑒𝑥−1 = 1 − 𝑥 2 + 𝐵1𝑥 2 2! − 𝐵2𝑥 4 4! + 𝐵3𝑥 6 6! −. .. 𝑥. 𝑐𝑜𝑡. 𝑥 = ∑ (−1)𝑛𝐵2𝑛(2𝑥) 2𝑛 (2𝑛)! ∞ 𝑛=0 1 − 𝑥 2 𝑐𝑜𝑡 𝑥 2 = 𝐵1𝑥 2 2! + 𝐵2𝑥 4 4! + 𝐵3𝑥 6 6! +. .. 𝐵2𝑛 = 2(−1)𝑛−1(2𝑛)! (2𝜋)2𝑛 (1 + 1 22𝑛 + 1 32𝑛 + 1 42𝑛 +. . . ) ( 2𝑛 + 1 2 )22𝐵1 − ( 2𝑛 + 1 4 ) 24𝐵2 + ( 2𝑛 + 1 6 )26𝐵3−. . . (−1) 𝑛−1(2𝑛 + 1)22𝑛𝐵𝑛 = 2𝑛 Bronshtein and Semendiaev [7] mention the other series that relate to the Bernoulli numbers: 𝐵𝑛 = 2(2𝑛)! (22𝑛−1)𝜋2𝑛 (1 + 1 32𝑛 + 1 52𝑛 + 1 72𝑛 +. . . ) 𝐵𝑛 = (2𝑛)! (22𝑛−1−1)𝜋2𝑛 (1 − 1 22𝑛 + 1 32𝑛 − 1 42𝑛 +. . . ) 𝐵𝑛 = (2𝑛)! 22𝑛−1𝜋2𝑛 (1 + 1 22𝑛 + 1 32𝑛 + 1 42𝑛 + 1 52𝑛 +. . . ) 𝐵𝑛 = 𝑛! × | | 1 2! 1 0 . . . 0 1 3! 1 2! 1 . . . 0 . . . . . . . 1 (𝑛+1)! 1 𝑛! 1 (𝑛−1)! . . . 1 2! | | Fernández [5], Varadarajan [20] and Bronshtein and Semendiaev [7] indicate that the expressions relating to Euler numbers are: 1 − 1 32𝑛+1 + 1 52𝑛+1 − 1 72𝑛+1 +. . . ± 1 (2𝑛−1)2𝑛+1 ∓. . . = 𝜋2𝑛+1 22𝑛+2×(2𝑛)! × 𝐸𝑛 𝑠𝑒𝑐. 𝑥 = 1 + 𝐸1.𝑥 2 2! + 𝐸2.𝑥 4 4! + 𝐸3.𝑥 6 6! +. .. 𝑠𝑒𝑐 ℎ . 𝑥 = 1 − 𝐸1.𝑥 2 2! + 𝐸2.𝑥 4 4! − 𝐸3.𝑥 6 6! +. .. 𝐸𝑛 = 22𝑛+2(2𝑛)! 𝜋2𝑛+1 (1 − 1 32𝑛+1 + 1 52𝑛+1 − 1 72𝑛+1 + 1 92𝑛+1 −. . . ) 𝐸𝑛 = | | | 1 2! 1 . . . 0 0 1 4! 1 2! . . . 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 . . . . . 1 2! 1 1 2𝑛! 1 2𝑛−2! . . . 1 4! 1 2! | | | 278 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate 𝐸𝑛 = | | | 1 1 0 . . . 0 1 ( 4 2 ) 1 . . . 0 1 ( 6 2 ) ( 6 4 ) . . . 0 . . . . . . . . . . . . . . . . . . . . . 1 ( 2𝑛 − 2 2 ) ( 2𝑛 − 2 4 ) . . . 1 1 ( 2𝑛 2 ) ( 2𝑛 4 ) . . . ( 2𝑛 2𝑛 − 2 ) | | | 2. Materials y Methods Methodology. The research has been divided into two parts. In the first part, we focused on calculating the first twenty Bernoulli numbers, and in the second part, we focused on calculating the first twenty Euler numbers. The most convenient, suitable, and important expression that relates Bernoulli numbers, according to Tsuneo et al. [21] and Fernández [5], which has been used in this study, is: ( 2𝑛 + 1 1 )𝐵𝑛 − ( 2𝑛 + 1 3 )𝐵𝑛−1 + ( 2𝑛 + 1 5 )𝐵𝑛−2 + (−1)𝑛−1 ( 2𝑛 + 1 2𝑛 − 1 )𝐵1 + (−1)𝑛 (𝑛 − 1 2 ) = 0, Procedure: Calculation of Bernoulli Numbers ( 2𝑛 + 1 1 )𝐵𝑛 − ( 2𝑛 + 1 3 )𝐵𝑛−1 + ( 2𝑛 + 1 5 )𝐵𝑛−2 + (−1)𝑛−1 ( 2𝑛 + 1 2𝑛 − 1 )𝐵1 + (−1)𝑛 (𝑛 − 1 2 ) = 0, The above expression has been used to evaluate the Bernoulli numbers (𝐵𝑛); whose calculations are presented according to the development outlined in summarized form below: Calculation, according to the binomial formula for the Bernoulli number, of 𝐵1 ( 3 1 )𝐵1 − (1 − 1 2 ) = 0; 3! 1!×2! 𝐵1 = 1 2 ⇒ 𝐵1 = 1 6 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵2 ( 5 1 )𝐵2 − ( 5 3 ) × 1 6 + (2 − 1 2 ) = 0; 5! 1!×4! 𝐵2 − 5! 3!×3! × 1 6 + 3 2 = 0 ⇒ 𝐵2 = 1 30 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵3 ( 7 1 ) × 𝐵3 − ( 7 3 ) × 1 30 + ( 7 5 ) × 1 6 − (3 − 1 2 ) = 0; 7! 1!×6! × 𝐵3 − 7! 3!×4! × 1 30 + 7! 5!×2! × 1 6 − 5 2 = 0 ⇒ 𝐵3 = 1 42 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵4 ( 9 1 ) × 𝐵4 − ( 9 3 ) × 1 42 + ( 9 5 ) × 1 30 − ( 9 7 ) × 1 6 + (4 − 1 2 ) = 0 9! 1!×8! × 𝐵4 − 9! 3!×6! × 1 42 + 9! 5!×4! × 1 30 − 9! 7!×2! × 1 6 + 7 2 = 0 ⇒ 𝐵4 = 1 30 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵5 279 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate ( 11 1 )𝐵5 − ( 11 3 ) 1 30 + ( 11 5 ) 1 42 − ( 11 7 ) 1 30 + ( 11 9 ) 1 6 − (5 − 1 2 ) = 0 11! 1!×10! 𝐵5 − 11! 3!×8! 1 30 + 11! 5!×6! × 1 42 − 11! 7!×4! 1 30 + 11! 9!×2! − 9 2 = 0 ⇒ 𝐵5 = 5 66 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵6 ( 13 1 )𝐵6 − ( 13 3 ) 5 66 + ( 13 5 ) 1 30 − ( 13 7 ) 1 42 + ( 13 9 ) 1 30 − ( 13 11 ) 1 6 + (6 − 1 2 ) = 0 13! 1!×12! 𝐵6 − 13! 3!×10! 5 66 + 13! 5!×8! 1 30 − 13! 7!×6! 1 42 + 13! 9!×4! 1 30 − 13! 11!×2! 1 6 + 11 2 = 0 ⇒ 𝐵6 = 691 2730 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵7 ( 15 1 )𝐵8 − ( 15 3 ) 691 2730 + ( 15 5 ) 5 66 − ( 15 7 ) 1 30 + ( 15 9 ) 1 42 − ( 15 11 ) 1 30 + ( 15 13 ) 1 6 − (7 − 1 2 ) = 0 15! 1!×14! 𝐵7 − 15! 3!×12! 691 2730 + 15! 5!×10! 5 66 − 15! 7!×8! 1 30 + 15! 9!×6! − 15! 11!×4! 1 30 + 15! 13!×2! 1 6 − 13 2 = 0 ⇒ 𝐵7 = 7 6 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵8 ( 17 1 )𝐵8 − ( 17 3 ) 7 6 + ( 17 5 ) 691 2730 − ( 17 7 ) 5 66 + ( 17 9 ) 1 30 − ( 17 11 ) 1 42 + ( 17 13 ) 1 30 − ( 17 15 ) 1 6 + (8 − 1 2 ) = 0 17! 1!×16! 𝐵8 − 17! 3!×14! 7 6 + 17! 5!×12! 691 2730 − 17! 7!×10! 5 66 + 17! 9!×8! 1 30 − 17! 11!×6! 1 42 + 17! 13!×4! 1 30 − 17! 15!×2! 1 6 + 15 2 = 0 ⇒ 𝐵8 = 3617 510 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵9 ( 19 1 )𝐵9 − ( 19 3 ) 3617 510 + ( 19 5 ) 7 6 − ( 19 7 ) 691 2730 + ( 19 9 ) 5 66 − ( 19 11 ) 1 30 + ( 19 13 ) 1 42 − ( 19 15 ) 1 30 + ( 19 17 ) 1 6 − (9 − 1 2 ) = 0 19! 1!×18! 𝐵9 − 19! 3!×16! 3617 510 + 19! 5!×14! 7 6 − 19! 7!×12! 691 2730 + 19! 9!×10! 5 66 − 19! 11!×8! 1 30 + 19! 13!×6! 1 42 − 19! 15!×4! 1 30 + 19! 17!×2! 1 6 − 17 2 = 0 ⇒ 𝐵9 = 43867 798 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵10 ( 21 1 )𝐵10 − ( 21 3 ) 43867 798 + ( 25 5 ) 3617 510 − ( 25 7 ) 7 6 + ( 21 9 ) 691 2730 − ( 21 11 ) 5 66 + ( 21 13 ) 1 30 − ( 21 15 ) 1 42 + +( 21 17 ) 1 30 − ( 21 19 ) 1 6 + (10 − 1 2 ) = 0 21! 1!×20! 𝐵10 − 21! 3!×18! 43867 798 + 21! 5!×16! 3617 510 − 21! 7!×14! 7 6 + 21! 9!×12! 691 2730 − 21! 11!×10! 5 66 + 21! 13!×8! 1 30 − 21! 15!×6! 1 42 + 21! 17!×4! 1 30 − 21! 19!×2! 1 6 + 19 2 = 0 ⇒ 𝐵10 = 174611 330 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵11 ( 23 1 )𝐵11 − ( 23 3 ) 174611 330 + ( 23 5 ) 43867 798 − ( 23 7 ) 3617 510 + ( 23 9 ) 7 6 − ( 23 11 ) 691 2730 + ( 23 13 ) 5 66 − ( 23 15 ) 1 30 + ( 23 17 ) 1 42 − ( 23 19 ) 1 30 + ( 23 21 ) 1 6 − (11 − 1 2 ) = 0 280 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate 23! 1!×22! 𝐵11 − 23! 3!×20! 174611 330 + 23! 5!×18! 43867 798 − 23! 7!×16! 3617 510 + 23! 9!×14! 7 6 − 23! 11!×12! 691 2730 + 23! 13!×10! 5 66 − 23! 15!×8! 1 30 + 23! 17!×6! 1 42 − 23! 19!×4! 1 30 + 23! 21!×2! 1 6 − 21 2 = 0 ⇒ 𝐵11 = 854513 138 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵12 ( 25 1 )𝐵12 − ( 25 3 ) 854513 138 + ( 25 5 ) 174611 330 − ( 25 7 ) 43867 798 + ( 25 9 ) 3617 510 − ( 25 11 ) 7 6 + +( 25 13 ) 691 2730 − ( 25 15 ) 5 66 + ( 25 17 ) 1 30 − ( 25 19 ) 1 42 + ( 25 21 ) 1 30 − ( 25 23 ) 1 6 + (12 − 1 2 ) = 0 25! 1!×24! 𝐵12 − 25! 3!×22! 854513 138 + 25! 5!×20! 174611 330 − 25! 7!×18! 43867 798 + 25! 9!×16! 3617 510 − 25! 11!×14! 7 6 + 25! 13!×12! 691 2730 − 25! 15!×10! 5 66 + 25! 17!×8! 1 30 − 25! 19!×6! 1 42 + 25! 21!×4! 1 30 − 25! 23!×2! 1 6 + 23 2 = 0 ⇒ 𝐵12 = 236364091 2730 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵13 ( 27 1 )𝐵13 − ( 27 3 ) 236364091 2730 + ( 27 5 ) 854513 138 − ( 27 7 ) 174611 330 + ( 27 9 ) 43867 798 − ( 27 11 ) 3617 510 + ( 27 13 ) 7 6 − ( 27 15 ) 691 2730 + ( 27 17 ) 5 66 − ( 27 19 ) 1 30 + ( 27 21 ) 1 42 − ( 27 23 ) 1 30 + ( 23 25 ) 1 6 − (13 − 1 2 ) = 0 27! 1!×26! 𝐵13 − 27! 3!×24! 236364091 2730 + 27! 5!×22! 854513 138 − 27! 7!×20! 174611 330 + 27! 9!×18! 43867 798 − 27! 11!×16! 3617 510 + 27! 13!×14! 7 6 − 27! 15!×12! 691 2730 + 27! 17!×10! 5 66 − 27! 19!×8! 1 30 + 27! 21!×6! 1 42 − 27! 23!×4! 1 30 + 27! 25!×2! 1 6 − 25 2 = 0 ⇒ 𝐵13 = 8553103 6 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵14 ( 29 1 )𝐵14 − ( 29 3 ) 8553103 6 + ( 29 5 ) 236364091 2730 − ( 29 7 ) 854513 138 + ( 29 9 ) 174611 330 − ( 29 11 ) 43867 798 + ( 29 13 ) 3617 510 − ( 29 15 ) 7 6 + ( 29 17 ) 691 2730 − ( 29 19 ) 5 66 + ( 29 21 ) 1 30 − ( 29 23 ) 1 42 + ( 29 25 ) 1 30 − ( 29 27 ) 1 6 + (14 − 1 2 ) = 0 29! 1!×28! 𝐵14 − 29! 3!×26! 8553103 6 + 29! 5!×24! 236364091 2730 − 29! 7!×22! 854513 138 + 29! 9!×20! 174611 330 − 29! 11!×18! 43867 798 + 29! 13!×16! 3617 510 − 29! 15!×14! 7 6 + 29! 17!×12! 691 2730 − 29! 19!×10! 5 66 + 29! 21!×8! 1 30 − 29! 23!×6! 1 42 + 29! 25!×4! 1 30 − 29! 27!×2! 1 6 + 27 2 = 0 ⇒ 𝐵14 = 23749461029 870 Calculation, according to the binomial formula for the Bernoulli number, of 𝐵15 ( 31 1 )𝐵15 − ( 31 3 ) 23749461029 870 + ( 31 5 ) 8553103 6 − ( 31 7 ) 236364091 2730 + ( 31 9 ) 854513 138 − ( 31 11 ) 174611 330 + ( 31 13 ) 43867 798 − ( 31 15 ) 3617 510 + ( 31 17 ) 7 6 − ( 31 19 ) 691 2730 + ( 31 21 ) 5 66 − ( 31 23 ) 1 30 + ( 31 25 ) 1 42 − ( 31 27 ) 1 30 + ( 31 29 ) 1 6 − (15 − 1 2 ) = 0 31! 1!×30! 𝐵15 − 31! 3!×28! 23749461029 870 + 31! 5!×26! 8553103 6 − 31! 7!×24! 236364091 2730 + 31! 9!×22! 854513 138 − 31! 11!×20! 174611 330 + 31! 13!×18! 43867 798 − 31! 15!×16! 3617 510 + 31! 17!×14! 7 6 − 31! 19!×12! 691 2730 + 31! 21!×10! 5 66 − 31! 23!×8! 1 30 + 31! 25!×6! 1 42 − 31! 27!×4! 1 30 + 31! 29!×2! 1 6 − 29 2 = 0 ⇒ 𝐵15 = 8615841276005 14322 281 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate With respect to the Euler numbers, the most important, convenient, and appropriate expression that relates Euler numbers according to Varadarajan [20] and Murray [22] and which was used to calculate the Euler numbers on this occasion, is: 𝐸𝑛 = ( 2𝑛 2 ) × 𝐸𝑛−1 − ( 2𝑛 4 ) × 𝐸𝑛−2 + ( 2𝑛 6 ) × 𝐸𝑛−3 − ( 2𝑛 8 )𝐸𝑛−4+. . . (−1) 𝑛 The above expression has been used to evaluate the Euler numbers (𝐸𝑛); whose calculations are presented in the development detailed below: Calculation, according to the binomial formula for the Euler number, of 𝐸1 𝐸1 = −(−1)1 ⇒ 𝐸1 = 1 Calculation, according to the binomial formula for the Euler number, of 𝐸2 𝐸2 = ( 4 2 ) × 1 − (−1)2 = 4! 2!×2! × 1 − 1 ⇒ 𝐸2 = 5 Calculation, according to the binomial formula for the Euler number, of 𝐸3 𝐸3 = ( 6 2 ) × 5 − ( 6 4 ) × 1 − (−1)3 = 6! 4!×21 × 5 − 6! 2!×4! × 1 + 1 ⇒ 𝐸3 = 61 Calculation, according to the binomial formula for the Euler number, of 𝐸4 𝐸4 = ( 8 2 )61 − ( 8 4 ) 5 + ( 8 6 ) 1 − (−1)4; 𝐸4 = 8! 6!×2! 61 − 8! 4!×4! 5 + 8! 2!×6! 1 − 1 ⇒ 𝐸4 = 1385 Calculation, according to the binomial formula for the Euler number, of 𝐸5 𝐸5 = ( 10 2 ) 1385 − ( 10 4 ) 61 + ( 10 6 ) 5 − ( 10 8 ) 1 + (−1)5 𝐸5 = 10! 8!×2! 1385 − 10! 6!×4! 61 + 10! 4!×6! 5 − 10! 2!×8! 1 + 1 𝐸5 = 50521 Calculation, according to the binomial formula for the Euler number, of 𝐸6 𝐸6 = ( 12 2 )50521 − ( 12 4 ) 1385 + ( 12 6 ) 61 − ( 12 8 ) 5 + ( 12 10 ) 1 − (−1)6 𝐸6 = 12! 10!×2! 50521 − 12! 8!×4! 1385 + 12! 6!×6! 61 − 12! 4!×8! 5 + 12! 2!×10! 1 − 1 𝐸6 = 2702765 Calculation, according to the binomial formula for the Euler number, of 𝐸7 𝐸7 = ( 14 2 ) 2702765 − ( 14 4 ) 50521 + ( 14 6 ) 1385 − ( 14 8 ) 61 + ( 14 10 ) 5 − ( 14 12 ) 1 − (−1)7 𝐸7 = 14! 12!×2! 2702765 − 14! 10!×4! 50521 + 14! 8!×6! 1385 − 14! 6!×8! 61 + 14! 4!×10! 5 − 14! 2!×12! + 1 ⇒ 𝐸7 = 199360981 Calculation, according to the binomial formula for the Euler number, of 𝐸8 𝐸8 = ( 16 2 ) 199360981 − ( 16 4 )2702765 + ( 16 6 ) 50521 − ( 16 8 )1385 + ( 16 10 )61 − ( 16 12 ) 5 + ( 16 14 )1 − (−1)8 𝐸8 = 16! 14!×2! 199360981 − 16! 12!×4! 2702765 + 16! 10!×6! 50521 − 16! 8!×8! 1385 + 16! 6!×10! 61 − 16! 4!×12! 5 + 16! 2!×14! − 1 ⇒ 𝐸8 = 19391512145 Calculation, according to the binomial formula for the Euler number, of 𝐸9 𝐸9 = ( 18 2 ) 19391512145 − ( 18 4 ) 199360981 + ( 18 6 ) 2702765 − ( 18 8 ) 50521 + ( 18 10 )1385 − ( 18 12 ) 61 + ( 18 14 ) 5 − ( 18 16 ) 1 − (−1)9 282 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate 𝐸9 = 18! 16!×2! 19391512145 − 18! 14!×4! 199360981 + 18! 12!×6! 2702765 − 18! 10!×8! 50521 + 18! 8!×10! 1385 − 18! 6!×12! 61 + 18! 4!×14! 5 − 18! 2!×16! 1 + 1 ⇒ 𝐸9 = 2404879675441 Calculation, according to the binomial formula for the Euler number, of 𝐸10 𝐸10 = ( 20 2 ) 2404879675441 − ( 20 4 ) 19391512145 + ( 20 6 ) 199360981 − ( 20 8 ) 2702765 + ( 20 10 ) 50521 − ( 20 12 )1385 + ( 20 14 )61 − ( 20 16 ) 5 + ( 20 18 )1 − (−1)10 𝐸10 = 20! 18!×2! 2404879675441 − 20! 16!×4! 1939151214 + 20! 14!×6! 199360981 − 20! 12!×8! 2702765 + 20! 10!×10! 50521 − 20! 8!×12! 1385 + 20! 6!×14! 61 − 20! 4!×16! 5 + 20! 2!×18! 1 − 1 ⇒ 𝐸10 = 370371188237525 Calculation, according to the binomial formula for the Euler number, of 𝐸11 𝐸11 = ( 22 2 )370371188237525 − ( 22 4 )2404879675441 + ( 22 6 ) 19391512145 − ( 22 8 ) 199360981 + ( 22 10 )2702765 − ( 22 12 ) 50521 + ( 22 14 ) 1385 − ( 22 16 ) 61 + ( 22 18 ) 5 − ( 22 20 ) 1 − (−1)11 𝐸11 = 22! 20!×2! 370371188237525 − 22! 18!×4! 2404879675441 + 22! 16!×6! 19391512145 − 22! 14!×8! 199360981 + 22! 12!×10! 2702765 − 22! 10!×12! 50521 + 22! 8!×14! 1385 − 22! 6!×16! 61 + 22! 4!×18! 5 − 22! 2!×20! 1 + 1 ⇒ 𝐸11 = 69348874393137901 Calculation, according to the binomial formula for the Euler number, of 𝐸12 𝐸12 = ( 24 2 )69348874393137901 − ( 24 4 ) 370371188237525 + ( 24 6 )2404879675441 − ( 24 8 ) 19391512145 + ( 24 10 ) 199360981 − ( 24 12 ) 2702765 + ( 24 14 ) 50521 − ( 24 16 )1385 + ( 24 18 )61 − ( 24 20 ) 5 + ( 24 22 )1 − (−1)12 𝐸12 = 24! 22!×2! 69348874393137901 − 24! 20!×4! 37037118823752 + 24! 18!×6! 2404879675441 − 24! 16!×8! 19391512145 + 24! 14!×10! 199360981 − 24! 12!×12! 2702765 + 24! 10!×14! 50521 − 24! 8!×16! 1385 + 24! 6!×18! 61 − 24! 4!×20! 5 + 24! 2!×22! 1 − 1 ⇒ 𝐸12 = 15514534163557086905 Calculation, according to the binomial formula for the Euler number, of 𝐸13 𝐸13 = ( 26 2 ) 15514534163557086905 − ( 26 4 ) 69348874393137901 + ( 26 6 )370371188237525 − ( 26 8 ) 2404879675441 + ( 26 10 )19391512145 − ( 26 12 ) 199360981 + ( 26 14 )2702765 − ( 26 16 )50521 + ( 26 18 ) 1385 − ( 26 20 ) 61 + ( 26 22 ) 5 − ( 26 24 ) 1 − (−1)13 𝐸13 = 26! 24!×2! 15514534163557086905 − 26! 22!×4! 69348874393137901 + 26! 20!×6! 370371188237525 − 26! 18!×8! 2404879675441 + 26! 16!×10! 19391512145 − 26! 14!×12! 199360981 + 26! 12!×14! 2702765 − 26! 10!×16! 50521 + 26! 8!×18! 1385 − 26! 6!×20! 61 + 26! 4!×22! 5 − 26! 2!×24! 1 + 1 ⇒ 𝐸13 = 4087072509293123892361 Calculation, according to the binomial formula for the Euler number, of 𝐸14 283 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate 𝐸14 = ( 28 2 )408707250929312892361 − ( 28 4 ) 15514534163557086905 + ( 28 6 ) 69348874393137901 − ( 28 8 )370371188237525 + ( 28 10 ) 2404879675441 − ( 28 12 ) 19391512145 + ( 28 14 ) 199360981 − ( 28 16 ) 2702765 + ( 28 18 ) 50521 − ( 28 20 )1385 + ( 28 22 )61 − ( 28 24 ) 5 + ( 28 26 )1 − (−1)14 𝐸14 = 28! 26!×2! 4087072509293123892361 − 28! 24!×4! 15514534163557086905 + 28! 22!×6! 69348874393137901 − 28! 20!×8! 370371188237525 + 28! 18!×10! 2404879675441 − 28! 16!×12! 19391512145 + 28! 14!×14! 199360981 − 28! 12!×16! 2702765 + 28! 10!×18! 50521 − 28! 8!×20! 1385 + 28! 6!×22! 61 − 28! 4!×24! 5 + 28! 2!×24! 1 − 1 ⇒ 𝐸14 = 1252259641403629865468285 Calculation, according to the binomial formula for the Euler number, of 𝐸15 𝐸15 = ( 30 2 )125225964140362865468285 − ( 30 4 )4087072509293123892361 + ( 30 6 ) 15514534163557086905 − ( 30 8 ) 69348874393137901 + ( 30 10 )370371188237525 − ( 30 12 ) 2404879675441 + ( 30 14 )19391512145 − ( 30 16 ) 199360981 + ( 30 18 )2702765 − ( 30 20 )50521 + ( 30 22 ) 1385 − ( 30 24 ) 61 + ( 30 26 ) 5 − ( 30 28 ) 1 − (−1)15 𝐸15 = 30! 28! × 2! 1252259641403629865468285 − 30! 26! × 4! 4087072509293123892361 + 30! 24! × 6! 15514534163557086905 − 30! 22! × 8! 69348874393137901 + 30! 20! × 10! 370371188237525 − 30! 18! × 12! 2404879675441 + 30! 16! × 14! 19391512145 − 30! 14! × 16! 199360981 + 30! 12! × 18! 2702765 − 30! 10! × 20! 50521 + 30! 8! × 22! 1385 − 30! 6! × 24! 61 + 30! 4! × 26! 5 − 30! 2! × 28! 1 + 1 ⇒ 𝐸15 = 441543893249023104553682821 3. Results Having carried out the appropriate and proposed calculations, we present the table of the first 20 Bernoulli and Euler numbers listed in Tables 1 and 2, in which the applications of Bernoulli and Euler numbers were carried out separately. The first application corresponds to the Bernoulli numbers, in solving the integral of ln⁡(𝑐𝑜𝑠𝑥). 284 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate Table 1. The Bernoulli Numbers. 𝒏 𝑩𝒏 1 1/6 2 1/30 3 1/42 4 1/30 5 5/66 6 691/2730 7 7/6 8 3617/510 9 43867/798 10 174611/330 11 854513/138 12 236364091/2730 13 8553103/6 14 23749461029/870 15 8615841276005/14322 16 7709321041217/510 17 2577687858367/6 18 26315271553053477373/1919190 19 2929993913841559/6 20 261082718496449122051/13530 3.1. For the Bernoulli numbers ∫ 𝑥 𝑠𝑒𝑛(𝑎𝑥) 𝑑𝑥 = 1 𝑎2 [𝑎𝑥 + (𝑎𝑥)3 3×3! + 7(𝑎𝑥)5 3×5×5! + 31(𝑎𝑥)7 3×7×7! +. . . + 2(22𝑛−1−1) (2𝑛+1)! 𝐵𝑛(𝑎𝑥) 2𝑛+1+. . . ] …(a) ∫𝑥𝑡𝑔(𝑎𝑥)𝑑𝑥 = 𝑎𝑥3 3 + 𝑎3𝑥5 15 + 2𝑎5𝑥7 105 + 17𝑎7𝑥9 2835 +. . . + 22𝑛(22𝑛−1)𝐵𝑛𝑎 2𝑛−1𝑥2𝑛+1 (2𝑛+1)! +. ..(b) ∫ 𝑡𝑔(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑎𝑥 + (𝑎𝑥)3 9 + 2(𝑎𝑥)5 75 + 17(𝑎𝑥)7 2205 +. . . + 22𝑛(22𝑛−1)𝐵𝑛(𝑎𝑥) 2𝑛−1 (2𝑛−1)(2𝑛)! +. ..(c) ∫ 𝑙𝑛( 𝑡𝑔𝑥)𝑑𝑥 = 𝑥 𝑙𝑛 𝑥 − 𝑥 + 𝑥3 9 + 7𝑥5 450 +. . . + 22𝑛(22𝑛−1−1)𝐵𝑛 𝑛(2𝑛+1)! 𝑥2𝑛+1+. . .(d) ∫ ln(𝑐𝑜𝑠𝑥) 𝑑𝑥 = − 22𝑛−1×(22𝑛−1)×𝐵𝑛×𝑥 2𝑛+1 𝑛(2𝑛+1)! ...(e) 3.1.1. Application of the Bernoulli Numbers (𝐵𝑛). For the expression ∫ ln(𝑐𝑜𝑠𝑥) 𝑑𝑥 (e); we have: • For: 𝐵1 = 1/6 ∫ln(𝑐𝑜𝑠𝑥) 𝑑𝑥 = − 𝑥3 6 • For: 𝐵1 = 1/6 y ⁡𝐵2 = 1/30 ∫ln(𝑐𝑜𝑠𝑥) 𝑑𝑥 = − 𝑥3 6 − 𝑥5 60 • For: 𝐵1 = 1/6,⁡𝐵2 = 1/30 y 𝐵3 = 1/42 ∫ln(𝑐𝑜𝑠𝑥) 𝑑𝑥 = − 𝑥3 6 − 𝑥5 60 − 𝑥7 210 • For: 𝐵1 = 1/6,⁡𝐵2 = 1/30, 𝐵3 = 1/42 y 𝐵4 = 1/30 ∫ ln(𝑐𝑜𝑠𝑥) 𝑑𝑥 = − 𝑥3 6 − 𝑥5 60 − 𝑥7 210 − 17𝑥9 22680 • For: 𝐵1 = 1/6,⁡𝐵2 = 1/30, 𝐵3 = 1/42, 𝐵4 = 1/30 y 𝐵5 = 5/66. 285 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate ∫ln(𝑐𝑜𝑠𝑥) 𝑑𝑥 = − 𝑥3 6 − 𝑥5 60 − 𝑥7 210 − 17𝑥9 22680 − 31𝑥11 311850 • For: 𝐵1 = 1/6,⁡𝐵2 = 1/30, 𝐵3 = 1/42, 𝐵4 = 1/30, 𝐵5 = 5/66 y 𝐵6 = 691/2730. ∫ ln(𝑐𝑜𝑠𝑥) 𝑑𝑥 = − 𝑥3 6 − 𝑥5 60 − 𝑥7 210 − 17𝑥9 22680 − 31𝑥11 311850 − 691𝑥13 4455 • For: 𝐵1 = 1/6,⁡𝐵2 = 1/30, 𝐵3 = 1/42, 𝐵4 = 1/30, 𝐵5 = 5/66, 𝐵6 = 691/2730 y 𝐵7 = 7/6. ∫ ln(𝑐𝑜𝑠𝑥) 𝑑𝑥 = − 𝑥3 6 − 𝑥5 60 − 𝑥7 210 − 17𝑥9 22680 − 31𝑥11 311850 − 691𝑥13 4455 − 10922𝑥15 638512875 • For: 𝐵1 = 1/6,⁡𝐵2 = 1/30, 𝐵3 = 1/42, 𝐵4 = 1/30, 𝐵5 = 5/66, 𝐵6 = 691/2730, 𝐵7 = 7/6 y 𝐵8 = 3617/510. ∫ ln(𝑐𝑜𝑠𝑥) 𝑑𝑥 = − 𝑥3 6 − 𝑥5 60 − 𝑥7 210 − 17𝑥9 22680 − 31𝑥11 311850 − 691𝑥13 4455 − 10922𝑥15 638512875 − 929569𝑥17 173675502000 +⋯; and so on, the Bernoulli numbers could continue to be applied to other integrals and mathematical expressions in which the Bernoulli numbers are necessary. The second application corresponds to the Euler numbers, in solving the integral of⁡[sec⁡(𝑎𝑥)] 𝑥⁄ . Table 2. The Euler numbers. n En 1 1 2 5 3 61 4 1385 5 50521 6 2702765 7 199360891 8 19391512145 9 2404879675441 10 370371188237525 11 69348874393137901 12 15514534163557086905 13 4087072509293123892361 14 1252259641403629865468285 15 441543893249023104553682821 16 177519391579539289436664789665 17 80723299235887898062168247453281 18 41222060339517702122347079671259045 19 23489580527043108252017828576198947741 20 14851150718114980017877156781405826684425 3.2. For the Euler Numbers ∫ 𝑥 𝑐𝑜𝑠(𝑎𝑥) 𝑑𝑥 = 1 𝑎2 [ (𝑎𝑥)2 2 + (𝑎𝑥)4 4×2! + 5(𝑎𝑥)6 6×4! + 61(𝑎𝑥)8 8×6! +. . . + 𝐸𝑛(𝑎𝑥) 2𝑛+2 (2𝑛+2)(2𝑛)! +. . . ]... (f) ∫𝑥 𝑠𝑒𝑐( 𝑎𝑥)𝑑𝑥 = 1 𝑎2 [ (𝑎𝑥)2 2 + (𝑎𝑥)4 8 + 5(𝑎𝑥)6 144 + 61(𝑎𝑥)8 5760 +. . . + 𝐸𝑛(𝑎𝑥) 2𝑛+2 (2𝑛+2)(2𝑛)! +. . . ]...(g) ∫ 𝑥 𝑐𝑜𝑠ℎ(𝑎𝑥) 𝑑𝑥 = 1 𝑎2 [ (𝑎𝑥)2 2 − (𝑎𝑥)4 8 + 5(𝑎𝑥)6 144 − 61(𝑎𝑥)8 5760 +. . . + (−1)𝑛𝐸𝑛(𝑎𝑥) 2𝑛+2 (2𝑛+2)(2𝑛)! +. . . ]...(h) ∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + 𝐸𝑛(𝑎𝑥) 2𝑛 2𝑛(2𝑛)! ...(i) 3.2.1. Application of the Euler Numbers (𝐸𝑛). For example, for the expression ∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 (i); we have: 286 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate For 𝐸1 = 1 ∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + (𝑎𝑥)2 4 For 𝐸1 = 1 y 𝐸2 = 5 ∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + (𝑎𝑥)2 4 + 5(𝑎𝑥)4 96 For 𝐸1 = 1, 𝐸2 = 5 y 𝐸3 = 61 ∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + (𝑎𝑥)2 4 + 5(𝑎𝑥)4 96 + 61(𝑎𝑥)6 4320 For 𝐸1 = 1, 𝐸2 = 5, 𝐸3 = 61 y 𝐸4 = 1385 ∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + (𝑎𝑥)2 4 + 5(𝑎𝑥)4 96 + 61(𝑎𝑥)6 4320 + 277(𝑎𝑥)8 64512 For 𝐸1 = 1, 𝐸2 = 5, 𝐸3 = 61, 𝐸4 = 1385 y 𝐸5 = 50521 ∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + (𝑎𝑥)2 4 + 5(𝑎𝑥)4 96 + 61(𝑎𝑥)6 4320 + 277(𝑎𝑥)8 64512 + 50521(𝑎𝑥)10 36288000 For 𝐸1 = 1, 𝐸2 = 5, 𝐸3 = 61, 𝐸4 = 1385, 𝐸5 = 50521 y 𝐸6 = 2702765 ∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + (𝑎𝑥)2 4 + 5(𝑎𝑥)4 96 + 61(𝑎𝑥)6 4320 + 277(𝑎𝑥)8 64512 + 50521(𝑎𝑥)10 36288000 + 540553(𝑎𝑥)12 1149603840 For 𝐸1 = 1, 𝐸2 = 5, 𝐸3 = 61, 𝐸4 = 1385, 𝐸5 = 50521, 𝐸6 = 2702765 y 𝐸7 = 199360891 ∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + (𝑎𝑥)2 4 + 5(𝑎𝑥)4 96 + 61(𝑎𝑥)6 4320 + 277(𝑎𝑥)8 64512 + 50521(𝑎𝑥)10 36288000 + 540553(𝑎𝑥)12 1149603840 + 199360891𝑥14 1220496076800 For 𝐸1 = 1, 𝐸2 = 5, 𝐸3 = 61, 𝐸4 = 1385, 𝐸5 = 50521, 𝐸6 = 2702765, 𝐸7 = 199360891 y 𝐸8 = 19391512145 ⁡⁡∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + (𝑎𝑥)2 4 + 5(𝑎𝑥)4 96 + 61(𝑎𝑥)6 4320 + 277(𝑎𝑥)8 64512 + 50521(𝑎𝑥)10 36288000 + 540553(𝑎𝑥)12 1149603840 + 199360891𝑥14 1220496076800 + 3878302429(𝑎𝑥)16 66952927641600 +⋯ For 𝐸1 = 1, 𝐸2 = 5, 𝐸3 = 61, 𝐸4 = 1385, 𝐸5 = 50521, 𝐸6 = 2702765, 𝐸7 = 199360891, 𝐸8 = 19391512145 y 𝐸9 = 2404879675441 ⁡⁡∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + (𝑎𝑥)2 4 + 5(𝑎𝑥)4 96 + 61(𝑎𝑥)6 4320 + 277(𝑎𝑥)8 64512 + 50521(𝑎𝑥)10 36288000 + 540553(𝑎𝑥)12 1149603840 + 199360891𝑥14 1220496076800 + 3878302429(𝑎𝑥)16 66952927641600 + 2404879675441(𝑎𝑥)18 115242726703104000 +… For 𝐸1 = 1, 𝐸2 = 5, 𝐸3 = 61, 𝐸4 = 1385, 𝐸5 = 50521, 𝐸6 = 2702765, 𝐸7 = 199360891, 𝐸8 = 19391512145, 𝐸9 = 2404879675441 y 𝐸10 = 370371188237525 ∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + (𝑎𝑥)2 4 + 5(𝑎𝑥)4 96 + 61(𝑎𝑥)6 4320 + 277(𝑎𝑥)8 64512 + 50521(𝑎𝑥)10 36288000 + 540553(𝑎𝑥)12 1149603840 + 199360891𝑥14 1220496076800 + 3878302429(𝑎𝑥)16 66952927641600 + 2404879675441(𝑎𝑥)18 115242726703104000 + 14814847529501(𝑎𝑥)20 1946321606541312000 +⋯ For 𝐸1 = 1, 𝐸2 = 5, 𝐸3 = 61, 𝐸4 = 1385, 𝐸5 = 50521, 𝐸6 = 2702765, 𝐸7 = 199360891, 𝐸8 = 19391512145, 𝐸9 = 2404879675441, 𝐸10 = 370371188237525 y 𝐸11 = 69348874393137901 ∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + (𝑎𝑥)2 4 + 5(𝑎𝑥)4 96 + 61(𝑎𝑥)6 4320 + 277(𝑎𝑥)8 64512 + 50521(𝑎𝑥)10 36288000 + 540553(𝑎𝑥)12 1149603840 + 199360891𝑥14 1220496076800 + 3878302429(𝑎𝑥)16 66952927641600 + 2404879675441(𝑎𝑥)18 115242726703104000 + 14814847529501(𝑎𝑥)20 1946321606541312000 + 69348874393137901(𝑎𝑥)22 24728016011107368960000 +⋯ For 𝐸1 = 1, 𝐸2 = 5, 𝐸3 = 61, 𝐸4 = 1385, 𝐸5 = 50521, 𝐸6 = 2702765, 𝐸7 = 199360891, 𝐸8 = 19391512145, 𝐸9 = 2404879675441, 𝐸10 = 370371188237525, 𝐸11 = 69348874393137901 y 𝐸12 = 15514534163557086905 ∫ 𝑠𝑒𝑐(𝑎𝑥) 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + (𝑎𝑥)2 4 + 5(𝑎𝑥)4 96 + 61(𝑎𝑥)6 4320 + 277(𝑎𝑥)8 64512 + 50521(𝑎𝑥)10 36288000 + 540553(𝑎𝑥)12 1149603840 + 199360891𝑥14 1220496076800 + 3878302429(𝑎𝑥)16 66952927641600 + 2404879675441(𝑎𝑥)18 115242726703104000 + 14814847529501(𝑎𝑥)20 1946321606541312000 + 69348874393137901(𝑎𝑥)22 24728016011107368960000 + 3102906832711417381(𝑎𝑥)24 2978152328319549308928000 +⋯ ; and so on, the Euler numbers could continue to be applied. 287 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate 4. Discussion Bernoulli numbers decrease until the third position and then increase indefinitely, coinciding with what Fernández [5] and Vera [6]; report; they are the smallest possible ratio between two relatively prime numbers, coinciding with what Benton mentions, [1] and that Bernoulli numbers have numerous applications in the field of applied mathematics, mathematical analysis, and engineering, such as derivatives and integrals, as mentioned by Ruíz [10]. When calculating Bernoulli numbers, these can be applied in a wide range of mathematical applications, initially to the concepts of definite and indefinite integrals of trigonometric, inverse, and logarithmic functions, and consequently in the mathematical modeling of a wide range of real cases involving transcendental function relationships; coinciding with what was mentioned by Bronshtein and Semendiaev [7]. The estimated Euler numbers can be used in the integral calculation of functions and expressions more complicated than those involving Bernoulli numbers. Therefore, their application and solution to different cases are more rigorous, enabling the solving of problems of greater difficulty and providing results that are closer to reality; coinciding with what was mentioned by Murray [22]. Euler numbers are all positive, agreeing with what Varadarajan [20] and Simmons [23] mention, at least from position one to position twenty. Like Bernoulli numbers, Euler numbers also participate in a large number of mathematical relationships and physical phenomena applied to technological, scientific, and engineering processes, such as those mentioned by Abramowitz and Stegun [4]. It is concluded that all the Euler numbers calculated are integers and positive, that the sum of two consecutive Euler numbers is divisible by 3 (three), that Euler numbers alternately end in 1 or 5, that Bernoulli numbers are all positive, that Bernoulli numbers, from the fourth position onward, grow indefinitely, that Bernoulli numbers are the smallest ratio between two relatively prime numbers, and there are other more important direct applications to the mathematics of Bernoulli and Euler numbers that appear in a wide variety of processes and that are truly the essence of engineering, applied mathematics, experimental models, and their various behaviors. Likewise, it is recommended, where possible, to design software and/or special computer programs to obtain Bernoulli and Euler numbers from twenty onward, in the manner indicated by the authors. Transparency: The authors confirm that the manuscript is an honest, accurate, and transparent account of the study; that no vital features of the study have been omitted; and that any discrepancies from the study as planned have been explained. This study followed all ethical practices during writing. Copyright: © 2025 by the authors. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). References [1] W. Benton, Encyclopedia britannica. USA: Tomo-III. Inc. Printed, 1964. [2] C. B. Boyer, History of mathematics (M. Martínez Pérez, Trans.). Madrid, Spain: Alianza Editorial, 1987. [3] R. Courant and F. John, Introduction to calculus and mathematical analysis: Volume 1 (Chapter 8, Trigonometric Series). México, D.F: Editorial Limusa, 2015. [4] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York: Dover Publications, Inc, 1965. [5] B. D. J. Fernández, "Bernoulli numbers: A study on their importance, consequences, and some applications in number theory," Thesis to Obtain the Degree of Bachelor of Science in Physics and Mathematics with a Specialization in Mathematics. National Polytechnic Institute, Higher School of Physics and Mathematics]. Mexico City, 2012. [6] F. Vera, Mathematics and numbers. Buenos Aires, Argentina: Kapelusz, 1960. [7] I. Bronshtein and R. Semendiaev, Mathematics handbook for engineers and students. Moscú, Rusia: Mir, 2018. [8] O. Marín, History of mathematics and its great exponents. Lima, Perú: VPA-UTP, 2003. https://creativecommons.org/licenses/by/4.0/ 288 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 274-288, 2025 DOI: 10.55214/2576-8484.v9i11.10856 © 2025 by the authors; licensee Learning Gate [9] J. M. Child, The early mathematical manuscripts of Leibniz. Chicago. Londres: The Open Court Publishing Company, 2017. [10] A. Ruíz, Number theory. Lima, Perú: San Marcos, 2016. [11] O. Martínez, M. Cand, F. Domínguez, and G. M. Forner Gambau, Historical background of the construction of logarithms. In The construction of logarithms: History and didactic project. México: Publicaciones de la Universidad de León, 2014. [12] F. Vega-Calderón, L. Gallegos-Cázares, and F. Flores-Camacho, "Conceptual difficulties in understanding Bernoulli's equation," Revista Eureka sobre Enseñanza y Divulgación de las Ciencias, vol. 14, no. 2, pp. 339-352, 2017. [13] Á. J. Sierra Cotes and L. A. Gómez Yance, "Determination of the influence of diameter, jet velocity and flow rate at different heights on the vertical discharge of fluids through orifices with inlet and outlet profiles," 2020. [14] F. Huancapaza, "Comparative analysis of numerical methods to obtain the dynamic response of structures," Undergraduate Thesis, Private University of Tacna. Tacna, Peru, 2021. [15] F. Torres, "Generation of a mathematical model for the study of reservoir operation with a multipurpose approach," Thesis for Obtaining the Degree of Mechanical Engineer in Fluid Mechanics. National University of San Marcos]. Lima, Peru, 2017. [16] R. R. Zamora, O. B. Manrique, and P. L. Balbón, "Energy balance in simple pipes applied to hydraulic engineering," Universidad & Ciencia, vol. 9, no. 2, pp. 2227–2690, 2020. [17] R. O. Estela Urbina et al., "Numerical methods applied to hydraulic calculation in irrigation canals of Bagua," Revista Científica Dékamu Agropec, vol. 3, no. 1, pp. 20-34, 2022. https://doi.org/10.55996/dekamuagropec.v3i1.70 [18] O. Marín-Machuca et al., "Quantification of the alkalinity of natural water by applying Bernoulli numbers and Simpson's approximate integral," BIOTEMPO, vol. 20, no. 2, pp. 183-196, 2023. https://doi.org/10.31381/biotempo.v20i2.5945 [19] J. Bernoulli, The art of guessing. Basel, Switzerland: Thurneysen, 1713. [20] V. Varadarajan, "Euler and his work on infinite series," Bulletin of the American Mathematical Society, vol. 44, no. 4, pp. 515-539, 2007. [21] A. Tsuneo, I. Tomoyoshi, and K. Masanobu, Bernoulli numbers and zeta functions (Springer Monographs in Mathematics; with an appendix by D. Zagier). Tokyo, Japan: Springer, 2014. [22] S. Murray, Manual of mathematical formulas and tables. New York: McGraw-Hill, Inc, 1970. [23] G. F. Simmons, Differential equations: With applications and historical notes, 2nd ed. Madrid, España: McGraw- Hill/Interamericana de España, S.A.U, 1998. https://doi.org/10.55996/dekamuagropec.v3i1.70 https://doi.org/10.31381/biotempo.v20i2.5945