When is integral divergent

First we compute the indefinite integral. Definition 2. Otherwise, we say the improper integral diverges. Evaluate it if it is convergent. With Example 2. For good measure, here is one more example. We use the Comparison Test to show that it converges. 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:. In this kind of integral one or both of the limits of integration are infinity.

In these cases, the interval of integration is said to be over an infinite interval. This is an innocent enough looking integral. This is a problem that we can do. So, this is how we will deal with these kinds of integrals in general. On a side note, notice that the area under a curve on an infinite interval was not infinity as we might have suspected it to be. In fact, it was a surprisingly small number. We will call these integrals convergent if the associated limit exists and is a finite number i.

Note as well that this requires BOTH of the integrals to be convergent in order for this integral to also be convergent. If either of the two integrals is divergent then so is this integral. We can actually extend this out to the following fact.

How fast is fast enough? The integral is then,. 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.