Calculation and application of the engineering numbers of Bernoulli (B_n) and Euler (E_n) to integral analysis and other applications

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 In (cos x) and sec(ax)/x; respectively.