Last week we finished with substitution and integration by parts. A practice test was handed out to help you prepare the test coming up next Tuesday/Wednesday. We will then start Chapter 6: Applications of integration.
Both substitution and integration by parts are techniques for finding antiderivatives, so the goal for both is to simplify the integrand so it fits into the table on page 392. To differentiate the two methods: substitution normally involves functions that look “non-basic”, i.e., functions that are composed of different functions; integration by parts normally involves product of two functions. Since you rarely see only substitution or IBP, it is a good idea to try first do a substitution and simplify your integrand. Then going from there, it might be clear to you whether to do more substitution or IBP. For both methods, remember to follow the general tips/orders carefully and patiently, you should be fine!
One thing I noticed: when you have decided on your u (both in substitution and IBP) and look for du, remember du means the “differential of u”, therefore you want to differentiate u!!. Your goal is to simplify the integrand, so if it looks like it is getting more complicated, check your du!
Next Thursday/Friday, we will start with applications of integrations, in particular, the definite integration. The key idea behind this chapter is the same idea we had for Riemann sum: cut things into little pieces and add them up one by one, take the limit as your cut as more and more fine and denote the limit as a definite integral. Once you have obtained a definite integral, you can go back to the three techniques we learned to compute the definite integral (see sum up of quiz 1). It is a very visual and fun chapter!
There is no written assignment due this week, but you are probably busy reviewing for the test (everything we have learned so far). If you want me to check your work, just take a picture of it and email it to me. The more work you show, the easier it will be for me to figure out the mistake/good idea you had.
Finally, remember to get some free points back by regrading!!
Alright, see you soon!