Problem Set 5—Integration by Parts (Mostly)

Purpose

This problem set is mainly intended to develop your ability to use integration by parts. It also reinforces your ability to use properties of trigonometric functions to evaluate integrals involving those functions.

Background

Section 8.2 of our textbook discusses integration by parts. We discussed it in class on February 10. Section 8.3 of the textbook discusses some tricks for integrating trigonometric functions, which I expect to discuss in class on February 11 and 12.

Activity

Solve each of the following problems:

Problem 1

Exercise 2 in section 8.2 of the textbook (integrate θ cos(πθ)).

Problem 2

Exercise 6 in section 8.2 of the textbook (evaluate the definite integral from 1 to e of x³ lnx).

Problem 3

Exercise 10 in section 8.2 of the textbook (integrate (x²-2x+1)e^2x).

Problem 4

Exercise 34 in section 8.2 of the textbook (integrate 1 / (x (lnx)²)).

Problem 5

Exercise 6 in section 8.3 of the textbook (integrate cos³(4x)).

Problem 6

Exercise 55 in section 8.2 of the textbook (find a volume of rotation based on the curve y = e^x). Note that our book calculates the general antiderivative of lnx in section 8.2, and you can use the result of this calculation without re-deriving it in your solution to this problem if you need to. Also note that the final answer to this problem is in the “Answers to Odd-Numbered Exercises” appendix at the back of the book, and you are welcome to use it to check your answer—but I will be more interested in seeing how you got the answer than in what its final value is when you turn your solution in.

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as reading through it will help me know what to focus on in the rest of the meeting.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.

My basic expectation in grading this exercise is that your solution will show that you understand how to solve each problem, although there may be arithmetic or copying mistakes, inefficient solution methods, incorrect or irrelevant statements incidental to the solution, or similar minor mistakes. If you understand how to solve all the problems and have no minor errors, I will consider the solution to be in between “what I expect” and “surprisingly beyond expectations.” I will consider solutions to be 3/4, 1/2, 1/4, or none of what I expect according to what (rough) fraction of the problems your solution shows understanding of, although I will raise grades slightly if it is clear by the end of your grading meeting that you have come to understand things you didn’t understand when you arrived at the meeting.