# Calculus

## 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 …

## Where Do Logarithms Come From?

Having answered many questions recently about logarithms, I realized we haven’t yet covered the basics of that topic. Here we’ll introduce the concept by way of its history, and subsequently we’ll explore how they work.

## Looks Like a Frustum, But …

Last time we looked at how to find the volume of a frustum of a pyramid or cone. But sometimes what looks at first like a rectangular frustum actually isn’t. This case turns out to have a more general formula almost as nice as what we have for an actual frustum. We’ll discover that the …

## Proving the Chain Rule: Details Matter

When we recently looked at the Chain Rule, I considered including two questions about its proof, but decided they would be too much. However, when a recent question asked about a different version of the same proof, I decided to post all three. It is a nice illustration of how a mathematician’s view of a …

## Implicit Differentiation: What to Do When It’s “Wrong”

(A new question of the week) Having just discussed the Chain Rule and the Product and Quotient Rules, a recent question about implicit differentiation (which we covered in depth two years ago) fits in nicely. This raises an important issue: when you get an apparently wrong answer, you may just have done something wise that …