Introduction & Chapter Objectives

Introduction

In many mathematical and physical problems, we encounter integrals that cannot be evaluated in the usual sense because the interval of integration is infinite or the integrand becomes unbounded.

Such integrals are called improper integrals

They extend the concept of definite integrals by using limits to assign a finite value whenever possible. 

In this chapter, we will study the different types of improper integrals, learn how to determine their convergence or divergence, and explore several techniques for their evaluation.

Improper integrals play a crucial role in various applications, such as probability theory, Fourier analysis, and physics, where infinite processes and unbounded functions frequently appear.

Chapter Objectives

At the end of this chapter, students should be able to:

   1. Define improper integrals of the first and second kind. 

    2.  Distinguish between different types of improper integrals

            (infinite limits and infinite    discontinuities). 

   3.  Evaluate improper integrals using limits. 

  4.    Determine the convergence or divergence of an improper integral. 

  5.    Apply convergence tests such as the comparison test and the limit comparison test. 

   6.  Use improper integrals in practical applications, such as probability and physics problems.