When do improper integrals diverge

To compute improper integrals, we use the concept of limits along with the Fundamental Theorem of Calculus. Since we are dealing with limits, we are interested in convergence and divergence of the improper integral. If the limit exists and is a finite number, we say the improper integral converges.

Otherwise, we say the improper integral diverges , which we capture in the following definition. First we compute the indefinite integral. Definition 2.

Otherwise, we say the improper integral diverges. Evaluate it if it is convergent. In this section we consider integrals where one or both of the above conditions do not hold. Such integrals are called improper integrals. An improper integral is said to converge if its corresponding limit exists; otherwise, it diverges.

The improper integral in part 3 converges if and only if both of its limits exist. It is not uncommon for the limits resulting from improper integrals to need this rule as demonstrated next. This integral will require the use of Integration by Parts. We have:. We have just considered definite integrals where the interval of integration was infinite. We now consider another type of improper integration, where the range of the integrand is infinite. Oftentimes we are interested in knowing simply whether or not an improper integral converges, and not necessarily the value of a convergent integral.

We provide here several tools that help determine the convergence or divergence of improper integrals without integrating. When does this limit converge -- i. Somehow the dashed line forms a dividing line between convergence and divergence.

These results are summarized in the following Key Idea. A basic technique in determining convergence of improper integrals is to compare an integrand whose convergence is unknown to an integrand whose convergence is known.

This is described in the following theorem. In order for the integral in the example to be convergent we will need BOTH of these to be convergent. If one or both are divergent then the whole integral will also be divergent. We know that the second integral is convergent by the fact given in the infinite interval portion above.

So, all we need to do is check the first integral. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width.

Example 1 Evaluate the following integral. Example 3 Determine if the following integral is convergent or divergent. If it is convergent find its value. Example 4 Determine if the following integral is convergent or divergent.

Example 5 Determine if the following integral is convergent or divergent. Example 6 Determine if the following integral is convergent or divergent. Example 7 Determine if the following integral is convergent or divergent.

Example 8 Determine if the following integral is convergent or divergent.