Improper Integrals - Calculus 2

In the last lesson we took a brief side trip and revisited limits from Calculus 1. We looked at a new technique known as L'Hopital's rule for evaluating limits involving indeterminate forms. Now while that may have seemed unrelated to our journey thus far through the various advanced integration techniques   of Calculus 2, it actually leads quite nicely into the final integration technique we need to cover.

This last integration technique deals with solving a special type of definite integral known as an improper integral. An improper integral can mean one of two things (or even both!):

1. The bounds/limits of integration for the definite integral are ∞ or -∞
2. The function we want to integrate is undefined for a value of x that lies between the bounds of integration, or is undefined for one of the bounds of integration (sometimes called infinite discontinuities)

Definite integrals with one or both of these features are improper integrals, and the process for solving them will look slightly different than what you are used to for past integrals. Since ∞ and -∞ are not finite values, we are unable to just simply "plug them into" the antiderivative like any other bounds when we evaluate the definite integral. In the same way, we cannot "plug in" undefined values of a function (infinite discontinuities) into its antiderivative. So what can we do?

The way we work around this issue is to introduce a limit to the integral (and there is the connection to the last lesson!). We will end up looking at the limit as the evaluation of the antiderivative approaches  ∞, -∞, or the value of an infinite discontinuity. Sometimes this will result in needing to use L'Hopitals Rule, but for now, how we set up these limits and then proceed with evaluating improper integrals is the focus of this lesson in Calculus 2. So let's get right into it!

In this lesson, you will learn:

• How to identify the two different types of Improper Integrals
• How to introduce a limit (or multiple limits) to improper integrals in order to evaluate them
• How to evaluate an improper integral once a limit has been introduced

I hope you find this video to be helpful!

-Josh