Improper Integrals

Integrating an improper integral

In general, when we talk about improper integrals, we’re talking about integrals where either one or both of the limits of integration are infinite. Nothing about the function you need to integrate is necessarily different than any other type of integral, it’s simply an issue of dealing with the infinite limits of integration. Luckily, that means that you can use all of the other integration tools you’ve learned to evaluate the integral, including u-substitution, integration by parts, partial fractions, etc.

Evaluating the limits of integration

Once you integrate the function, you’ll be left to deal with the limits of integration. As with all other definite integrals you’ve seen before, you need to plug in the upper limit of integration first, and then subtract from that what you get when you plug in the lower limit of integration. The problem with an improper integral is that, technically, you can’t just “plug in” an infinite limit of integration. Instead, what you’ll generally do to deal with improper integrals is to substitute a variable for the infinite limit of integration, and then evaluate the result as the limit of that variable approaches infinity.

Convergent and Divergent improper integrals

If the limit exists and is a finite number (not positive or negative infinity), the we’ll call the improper integral convergent. Otherwise, the limit will either not exist, or be equal to positive or negative infinity, in which case we’ll call the improper integral divergent.