Why improper integral

So, the first integral is convergent. Note that this does NOT mean that the second integral will also be convergent. This integral is convergent and so since they are both convergent the integral we were actually asked to deal with is also convergent and its value is,.

In most examples in a Calculus II class that are worked over infinite intervals the limit either exists or is infinite. These are integrals that have discontinuous integrands. The process here is basically the same with one subtle difference. Note as well that we do need to use a left-hand limit here since the interval of integration is entirely on the left side of the upper limit.

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. Determine whether the following improper integrals are convergent or divergent. Evaluate those that are convergent. We first compute the corresponding indefinite integral using trigonometric substitution. We first solve the corresponding indefinite integral using the following substitution:. We now notice that the integral appears on both sides of the equation, and so we can combine terms:.

We know that. Section 2. Solution We use the Comparison Test to show that it converges. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. 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.