Convergence and Computation of a Class of Generalized Integrals

TL;DR

This paper investigates the convergence properties of a class of generalized integrals involving sine functions and power functions, deriving unified formulas and criteria using various mathematical methods.

Contribution

It introduces a unified approach to compute and analyze the convergence of generalized integrals of the form ∫₀^∞ (sin^b x)/x^a dx, expanding existing mathematical frameworks.

Findings

Derived unified calculation formulas for the integrals.

Established convergence and divergence criteria.

Applied multiple mathematical methods for derivation.

Abstract

In this study, we discuss the convergence and divergence of generalized integrals,\int_{0}^{+\infty}\frac{sin^{b}x}{x^{a}}dx(a\epsilon R^{+},b\epsilon N^{+}), and use the transformation method, the partial integration method, the mathematical induction method and the complex variable residual method to derive the calculation formulas of the two types of integrals. The calculation formulas of the two types of integrals are unified. Finally, the convergence criterion and the unified calculation formula of generalized integrals of\int_{0}^{+\infty}\frac{sin^{b}x}{x^{a}}dxare obtained.