The Indefinite Integral, and Techniques of Integration (I: Direct Methods and Substitution)
In Week 2 we discussed how, in order to compute definite integrals (areas of regions determined by graphs of functions), we need to be able to find antiderivatives for given functions. This is a consequence of the Fundamental Theorem of Calculus.
This week we will meet the concept of the indefinite integral of the expression f(x), which is the general antiderivative of f(x); it takes the form F(x)+C where F is a function satisfying F'(x)=f(x). Unfortunately there is no simple or uniform way to reverse all the rules of differentiation, but there are techniques available that can be applied to particular classes of examples. For integration tasks that can’t be tackled directly, one method that we will discuss this week is substitution, which is a means of reversing the chain rule of differentiation. It is a technique rather than a theorem, and we will study it via examples.
The relevant section of the Lecture notes this week is the first half of Section 1.4. We will look at the rest of Section 1.4 in Week 4. There are more examples in the lecture notes than we will discuss in our lectures. In Lecture 5, we will look at some examples that can be managed in a direct way, without recourse to any particular technique, and we will have a first look at the substitution approach.
Slides for this week.
Here are (old editions of) Lectures 5 and 6.
Supplementary video tutorial with more substitution examples
Notes from this tutorial.
Weekly Problem 3
The weekly problems are just for fun. They have nothing much to do with our curriculum. Please send me an email if you have a solution that you would like to share with the class!