To understand whether an improper integral converges, you need to know what kind of integral you are dealing with. There are two main situations where you might have an improper integral:

1. Infinite Limits of Integration: This happens when one or both ends of the integral stretch out to infinity. For example:

   • $\int_{a}^{\infty} f(x) \, dx$
   • $\int_{-\infty}^{\infty} f(x) \, dx$

2. Discontinuities in the Range: This occurs when the function $f(x)$ isn’t defined for some values between $a$ and $b$. For instance:

   • $\int_{a}^{b} f(x) \, dx$ where $f(x)$ is undefined at some point in the range $(a, b)$.

To check if the integral converges (meaning it approaches a specific number), you can follow these steps:

1. Limit Approach

For integrals with infinite limits, we can handle the infinity by using a limit.

For example, for the integral $\int_{a}^{\infty} f(x) \, dx,$ you rewrite it as: $\lim_{t \to \infty} \int_{a}^{t} f(x) \, dx.$

2. Checking Discontinuities

If there are points where the function is undefined, you should break the integral into parts. For example, if $f(x)$ isn't defined at $c$, you can split it like this:

$\lim_{t \to c^-} \int_{a}^{t} f(x) \, dx + \lim_{s \to c^+} \int_{s}^{b} f(x) \, dx.$

3. Evaluate the Limits

Now, calculate the limits you just set up.

If one or both of these limits result in a finite number, the improper integral converges. But if they go to infinity or don’t exist, then the integral diverges.

Comparison Test

Sometimes, it helps to compare the function you’re looking at with another function. Here’s how it works:

• If $f(x) \geq 0$ and you find another function $g(x)$ such that:
  • The integral $\int_{a}^{\infty} g(x) \, dx$ diverges, and since $f(x) \geq g(x)$, it means that $\int_{a}^{\infty} f(x) \, dx$ must also diverge.
• On the other hand, if $\int_{a}^{\infty} g(x) \, dx$ converges and $f(x) \leq g(x)$, then $\int_{a}^{\infty} f(x) \, dx$ converges.

In Summary

You can find out if an improper integral converges by carefully following the steps to handle limits, check for points where the function is not defined, and use comparison tests. This method will help you determine whether the integral approaches a specific value or not.