Improper Integrals: Survey of Methods

The first thing to remember is that any definite integral may be an improper integral - it is not always clear at the first sight. Whenever we are given a definite integral to evaluate, we have to scan the integration interval for possible troubles. There are two kinds of trouble:

If there are no troubles, we can evaluate the integral directly, most likely using an antiderivative (see Evaluating definite integrals in Methods Survey).

If there are any troubles, we have to keep in mind that we can safely evaluate only improper integrals that have just one problem, namely at one of the endpoints. In case we have more problems, we have to split the integration interval into smaller intervals so that each contains at most one trouble (at some endpoint). For the whole integral to converge, all smaller parts must converge.

It is therefore enough to know how to handle two basic types of improper integrals: those with trouble at the right endpoint and those with trouble at the left endpoint, with no other trouble present. There are also two basic questions concerning such integrals: evaluation and deciding on convergence.

Evaluating Improper Integrals

The question is simple: Evaluate (if it converges)

Solution is based on the definition of an improper integral.

If there is a trouble at the right endpoint b - typically we would also require that f be continuous on [a,b) - we cut the trouble away in this way:

If there is a trouble at the left endpoint a - typically we would also require that f be continuous on (a,b] - we cut the trouble away in this way:

The integral converges if the limit converges, we then get an answer. Otherwise the improper integral diverges (but sometimes it still makes sense to assign an answer, infinity or negative infinity).

In calculations we usually apply limit to the problem endpoint at the end, after evaluating the indefinite integral. In other words, we treat both integration endpoints, the problematic one and the other one, in the usual way including changing them in substitution if applicable, and only at the moment of substituting them (and perhaps when changing it in substitution) we use the limit procedure.

When evaluating the limit, it is important to apply it to the whole antiderivative, not to split it into several limits unless we are sure that it would not lead to trouble. To put it another way, whenever possible, express the antiderivative as one term, not as a sum of more.


Testing Convergence of Improper Integrals

The question is simple: Decide whether the following integral converges:

There are essentially three methods to answer this question:

1. Evaluate the integral. Naturally, this we do only if the evaluation is very easy or if the other two methods fail.

2. Use the Comparison test.

First, find a test function g which is either always greater than f or always smaller than f - here "always" means on the integration interval or at least on some appropriate neighborhood of the "problem". Typically, the test function is related to f but is "nicer", that is, easier to integrate. Second, investigate the convergence of the integral of g from a to b. Third, make a conclusion concerning the given integral.

Since the conclusion of this test is not in the form of equivalence, it is a "one-way" test and only works in some situations - namely, if the inequality between f and the test function allows us to reach some conclusion regarding their integrals. Sometimes the only possible conclusion is that there is no conclusion possible. This means that there are two possible approaches that would make this test successful:

3. Use the Limit Comparison test.

First, find a test function g which is (up to a non-zero multiple) about the same as f when x is close to the problem point. Second, justify your choice of the test function by finding the limit of the ratio f /g as x approaches the problem point (from an appropriate side). This limit should exist and not be zero. If this works, third, investigate the convergence of the integral of g from a to b. This conclusion then also applies to the given integral.

How to find the "right" test function:

The basic idea is that we look at the expression defining f and try to ignore all parts that we think are not important when x is near the problem. If we do it correctly, we should end up with a test function that is simpler, similar to f near the problem (which allows us to use the Limit Comparison test), and if we are lucky, this test function may also be always greater (or always smaller) than f, which would allow us to use the simpler Comparison test - of course, then we would have to be lucky again to get a conclusion, that is, we would hope that the inequality goes the right way.

Since this procedure most often yields a power (or rather, a reciprocal power), it is important that we remember the scale of powers at infinity and zero - see Theory - Improper Integrals - Examples.

This procedure seems easier if we have the problem of infinite integral limit. For instance, assume that in our integral b = ∞. Then we look at all sums in the expression for f and keep in each of them just the dominant term. From our experience with limits we know that when x is close to infinity, exponentials beat powers, these in turn beat powers of logarithms. When given a choice between different powers, the highest wins. Typically, given a fraction, we would pick the dominant terms in the numerator and the denominator and obtain the best test function.

If the problem is caused by lack of continuity, this may be more difficult and we have to apply a different procedure. Typically, we get such a problem when we have a fraction and the denominator happens to be zero at the considered point. In this case we have to use a different strategy than above for creating a test function. Whereas around infinity we ignore "unimportant parts" in sums, here we should ignore unimportant terms of products. That is, we express f as a product of terms as simple as possible (some terms will be in the denominator) and then keep only those terms that are equal to zero at the problem point. Another popular tool is a suitable Taylor expansion centered at the problem point, which can be used to substitute polynomial for an unpleasant function.