Which integrals are improper

In this case we need to use a right-hand limit here since the interval of integration is entirely on the right side of the lower limit.

As with the infinite interval case this requires BOTH of the integrals to be convergent in order for this integral to also be convergent. Again, this requires BOTH of the integrals to be convergent in order for this integral to also be convergent. Note that the limits in these cases really do need to be right or left-handed limits. Since we will be working inside the interval of integration we will need to make sure that we stay inside that interval.

One of the integrals is divergent that means the integral that we were asked to look at is divergent. Consider the following integral. This is an integral over an infinite interval that also contains a discontinuous integrand.

It is important to remember that all of the processes we are working with in this section so that each integral only contains one problem point. 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. Wolfram Education Portal » Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.

Wolfram Language » Knowledge-based programming for everyone. Terms of Use. Improper Integrals. For example, the integral. Mathematica » The 1 tool for creating Demonstrations and anything technical.

Contact the MathWorld Team. How to recognize and evaluate improper integrals when the interval of integration is finite? Ask Question. Asked 7 years, 3 months ago.

Active 7 years, 3 months ago. Viewed 7k times. Mariana A. Bomfim Mariana A. Bomfim 11 1 1 silver badge 2 2 bronze badges. Add a comment. Active Oldest Votes. I hope this helps clear up what an improper integral is!

BeaumontTaz BeaumontTaz 2, 2 2 gold badges 10 10 silver badges 14 14 bronze badges. 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.