Extension: Testing Convergence of Improper Integrals Using the Definition

By now you should be familiar with the most straightforward method of determining whether an "improper" integral converges: simply evaluating the limit directly. This method can be applied to both types of "improper" integral, those with endpoints at $\infty$ or $-\infty$ and those with interior discontinuities.

As $\int \frac{1}{x} dx = \ln(x) + C$, we can evaluate the integral as

$\lim_{t\to\infty} \int_{1}^{t} \frac{1}{x} dx = \lim_{t\to\infty}(\ln(t) - \ln(1)) = \lim_{t\to\infty} \ln(t)$.

The limit does not exist, so the integral diverges.

This method, however, cannot determine whether an integral converges every time. Other, more comprehensive methods will be discussed in the next few pages.

The definite integral is the area under the graph of a function within two boundaries. The standard notation for a definite integral is

$\int_a^{b}f(x)dx$

Sometimes an integral will have discontinuous endpoints; this is an improper integral and will take a form similar to that of the following example:

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

While this may look like a fairly normal function, we cannot just plug $\infty$ into $f(x)$ to get the solution for the limit.

The way that we solve this problem is by using the definition of the limit. There are several other ways to solve improper integrals that also use the limit definition, some of which can be found here: Bad Finite Endpoints, Multiple Singularities and Infinite Endpoints.

We begin solving our problem by replacing the discontinuous endpoint with an arbitrary, but real, endpoint; let's call this point $t$.

Then we change the limit to $\lim_{t\to\infty}$ so that instead of having the endpoint $\infty$, we set $t$ to approach $\infty$; $t$ will never actually equal $\infty$, it will just get extremely close, so that we are still able to plug it in once we integrate the function.

Now made into a proper function, it looks like

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