Techniques of Integration (II and III: Integration by Parts, and Partial Fractions
Welcome to Week 4. Our business this week is to discuss two more techniques for finding antiderivatives, and hence for calculating definite integrals. On Wednesday we will look at the Integration by Parts technique. This is a really important one and is essentially the best version we have of a reversal of the product rule of differentiation. It applies to situations where the integral involves two factors, at least one of which becomes more manageable when it is differentiated. That is not a very definitive description, but our collection of techniques is like a toolbox, with various implements that can be applied to different problems. A sense of how the tools work and which ones might apply to a given situation can be developed with experience.
On Thursday we will look at the partial fraction expansion technique. This is a useful arithmetic method that turns up from time to time, not only in integration problems. Basically it is the “opposite” of the method of adding fractions by finding a common denominator, that we all learned to do at a much earlier stage of education. If you have a fraction whose denominator is a product of different factors, a partial fraction expansion rewrites it as a sum of separate fractions with the various factors as denominators. These are generally easier to integrate than the version you started with.
Relevant sections of the Lecture notes this week are Sections 1.4.2 (from page 23) and Section 1.4.3 (from page 26). We will look at the final part of Chapter 1 in Week 5. There are more examples in the lecture notes than we will discuss in our lectures.
Here is (last year’s edition of) Lecture 7, on integration by parts.
Slides for Lecture 7: without annotation, and annotated.
Slides for Lecture 8 without annotation, and annotated.
Supplementary video tutorial with a few more examples
Some more “typical” examples, similar to those in the first three problems of Homework sheet 2. Notes from this tutorial.
Weekly Problem 4
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!