Solve Improper Integrals with One or Two Infinite Limits of Integration

By Mark Ryan

When improper integrals have one or two infinite limits of integration, you can solve these integrals by turning them into limits where c approaches infinity or negative infinity. Here are two examples:


So this improper integral converges.

In the next integral, the denominator is smaller, x instead of x2, and thus the fraction is bigger, so you’d expect


which it is. But it’s not just bigger, it’s way bigger:


This improper integral diverges.

Two functions.
Two functions.

This figure shows these two functions: the area under


The area under


is exactly the same as the area of the 1-by-1 square to its left: 1 square unit. The area under


is much, much bigger — actually, it’s infinitely bigger than a square large enough to enclose the Milky Way Galaxy. Their shapes are quite similar, but their areas couldn’t be more different.

When both of the limits of integration are infinite, you split the integral in two and turn each part into a limit. Splitting up the integral at x = 0 is convenient because zero’s an easy number to deal with, but you can split it up anywhere you like. Zero may also seem like a good choice because it looks like it’s in the middle between


But that’s an illusion because there is no middle between


or you could say that any point on the x-axis is the middle.

Here’s an example:


  1. Split the integral in two.


  2. Turn each part into a limit.


  3. Evaluate each part and add up the results.


You might want to do this problem again, splitting up the integral somewhere other than at x = 0, to confirm that you get the same result.

If either “half” integral diverges, the whole, original integral diverges.