Calculation of Generalized Dirichlet Integral: From Special Cases to General Formula

The Dirichlet integral is widely used in the fields of mathematical analysis, probability theory and physics. This paper explores the calculation of the generalized Dirichlet integral from zero to infinity. The author focuses on transitioning from special cases to deriving a general formula. The methodology used mainly include substitution and integration by parts. In the simplification of formulas, trigonometric identities and Frullani integral are also used. Moreover, the author obtains the general formula by discussing the odd and even power cases respectively. This paper deduces the general formula of Dirichlet integral by using Eulers formula and binomial expansion. The result demonstrates that it uses special cases to find a general formula with the different order power and even the particular case of it, which is the same order power. The formula simplifies calculations. The significance of this paper lies in the calculation of Dirichlet integral general formula and various variations and give the answer. It provides an accurate formula for other studies using the this integral, and enhancing the overall body of knowledge in integral calculus.