Sign Issues in Integration
Several recent questions involve things that go wrong with signs in integrating, and reveal some subtleties that are easily overlooked. We’ll also see some creative thinking!
Integration
Two-sided Improper Integrals: Can I Take Both Limits at Once?
We have a question about an improper integral, where one is strongly tempted to take a shortcut that makes it convergent, though the proper definition does not. Why can’t we do this? We’ll see something of the freedom mathematicians have in the matter of definitions, as well as why the standard definition has to be …
Two-sided Improper Integrals: Can I Take Both Limits at Once? Read More »
Integration: Sometimes It Just Can’t Be Done!
Having looked at what it takes to work out an indefinite integral, using all our tools, we need to face something that isn’t explained often enough: Some integrals aren’t just difficult; they’re impossible! We’ll look at what we’ve said in several cases where this issue arose.
Integration: It Takes a Whole Toolbox
Individual techniques of integration, as discussed in the last two posts, don’t represent the reality of the process, any more than demonstrating how to use a hammer or a screwdriver shows how to do carpentry. Let’s look at two questions we’ve had about challenging integrals that require a combination of methods. We’ll be using substitution, …
Integration: Choosing a Substitution to Try
Having looked at two basic techniques of integration, let’s start putting things together. How do you approach an integral without knowing what method to use? We’ll focus on substitution here, which is also called “change of variables”.
Figuring Out Integration by Parts
Having looked at some issues in integration, let’s look at some old questions about integration by parts.
Integrating Rational Functions: Beyond Partial Fractions
A couple recent questions offered tricks for integrating rational functions, opportunistically modifying or working around the usual method of partial fractions. We have previously discussed this method in Partial Fractions: How and Why, and in Integration: Partial Fractions and Substitution, where we looked at other variations.
Using Symmetry to Find a Definite Integral
(A new question of the week) Definite integrals can sometimes be solved by finding an antiderivative; but when that is either difficult or impossible, there may be special tricks available. Here we’ll lead a student gradually to a solution using symmetry; and then we’ll look at an earlier problem that used essentially the same trick …
Writing a Riemann Sum as an Integral
(A new question of the week) Riemann sums are used in defining the definite integral. But they can also be used in reverse: Sometimes you can be given the limit of a summation and asked to read it as a Riemann sum, and then turn it into an integral. Usually this is fairly straightforward; but …