When an

**integral diverges**, it fails to settle on a certain number or it’s value is ±infinity. More formally, we say that a divergent integral is where an improper integral’s limit doesn’t exist. On the other hand, if the limit is finite

*and*that limit is the value of the improper integral, the integral is convergent [1].

To put this in plain English, the term “integral diverges” means that you can’t calculate the area under a curve. For example, the following graph shows the function f(x) = 1/x:

Let’s say you wanted to find the area from x = 1 to x = ∞ (your bounds of integration). You can’t calculate a number because as the graph travels to infinity (blue arrow), the area will constantly increase a little more. When the **trouble spot** is at the end of the graph (more formally called an endpoint), it’s called a **simple improper integral** [2].

## How to Show if an Integral Diverges

Improper integrals aren’t that useful in calculus; When you come across one, the first step is usually to replace it with a proper integral (one with defined limits of integration).

In the above image, the improper integral on the left can’t be evaluated. To find out if the integral diverges:

- Replace the improper integral with a limit of a proper integrals.
- Find the limit (if it exists).

If the limit doesn’t exist, the integral diverges [3].

## Examples

**Example problem #1**: Does the following improper integral diverge?

Step 1: Replace the improper integral with a limit of a proper integrals:

Step 2: Find the limit:

The limit is a finite number (1), so this integral converges.

**Example problem #2**: Does the following improper integral diverge?

Step 1: Replace the improper integral with a limit of a proper integrals:

Step 2: Find the limit:

The limit is infinite, so this integral diverges.

The integral test is used to see if the integral converges; It also applies to series as well. If the test shows that the improper integral (or series) doesn’t converge, then it diverges.

## References

Graph: Desmos.com.

[1] Blair, R. Improper Integrals (With Solutions). Retrieved April 7, 2021 from: https://www2.math.upenn.edu/~ryblair/Math104/papers/Lec3_12Sol.pdf

[2] Swenton, F. (2009). Overview of Improper Integrals. Retrieved April 7, 2021 from: https://web.math.princeton.edu/~nelson/104/ImproperIntegrals.pdf

[3] Wakefield, N. et al. 5.10 Improper Integrals. Retrieved April 7, 2021 from: https://mathbooks.unl.edu/Calculus/sec-5-10-improper.html