Improper integrals are a type of math that can deal with infinity. When we want to find the value of an integral that goes to infinity, we need a clear way to check if it works or not.

You might see these types of integrals written like this:

$$\int_a^{\infty} f(x) \, dx$$

or

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

To make sense of these integrals, we can rewrite them using limits.

For the integral from a value 'a' to infinity, we can say:

$$\int_a^{\infty} f(x) \, dx = \lim_{t \to \infty} \int_a^t f(x) \, dx$$

And for the integral that starts from negative infinity, we write:

$$\int_{-\infty}^{b} f(x) \, dx = \lim_{t \to -\infty} \int_t^b f(x) \, dx$$

Using limits like this helps us manage the infinite parts of the integrals. By first calculating the definite integrals (where the limits aren’t infinite), we can learn how the function ( f(x) ) behaves as it gets close to the boundary.

When Do Improper Integrals Converge?

1. Going to Zero: First, we need to check that ( f(x) ) gets closer to ( 0 ) as ( x ) gets really big or really small (goes to negative infinity). If it doesn’t, the integral might diverge, which means it doesn’t work.

2. Comparison Test: Another way to check is by comparing ( f(x) ) to another function we already know. If we find a function ( g(x) ) that is bigger than ( f(x) ) (like ( 0 \leq f(x) \leq g(x) )) and if the integral of ( g(x) ) converges, then we can say that the integral of ( f(x) ) also converges.

Example

A simple example is the integral

$$\int_1^{\infty} \frac{1}{x^2} \, dx.$$

We rewrite it using limits:

$$\lim_{t \to \infty} \int_1^t \frac{1}{x^2} \, dx = \lim_{t \to \infty} \left[-\frac{1}{x} \right]_1^t = \lim_{t \to \infty} \left(-\frac{1}{t} + 1\right) = 1.$$

In this case, the integral converges, and the answer is ( 1 ).

Using this step-by-step method shows how to carefully work through improper integrals with infinite limits, helping us better understand if they give us a valid answer.