Convergence and Computation of a Class of Generalized Integrals

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.