Improper integrals are a special type of math problem. They come up when we deal with limits that head towards infinity or when we have functions that aren’t steady. Here are a couple of common situations where they occur:

1. Infinite Intervals:

   • An example is the integral from 1 to infinity of (\frac{1}{x^2}).
   • Another example is the integral from 0 to infinity of (e^{-x}).

2. Discontinuities:

   • We can also see this with the integral from 0 to 1 of (\frac{1}{\sqrt{x}}).
   • Or with the integral from -1 to 1 of (\frac{1}{x}).

To figure out if these integrals converge (meaning they settle on a specific value), we compare them to functions that we know converge.

We use methods like the Comparison Test or the Limit Comparison Test.

For example, when we look at (\int_{1}^{\infty} \frac{1}{x^2} , dx), we find that it converges to 1. On the other hand, (\int_{-1}^{1} \frac{1}{x} , dx) does not settle down to a particular value; we say it diverges.