Problem Set 6—Trigonometric Substitutions and Partial Fractions

Purpose

This problem set reinforces your ability to use trigonometric substitutions and the method of partial fractions to evaluate integrals.

Background

Section 8.4 of our textbook discusses trigonometric substitions. We discussed it in class on February 17. Section 8.5 of the textbook discusses the method of partial fractions. We will talk about this in class on February 18 and 19.

Activity

Solve each of the following problems:

Problem 1

Exercise 132 in section 7.6 of the textbook (compare the functions y = (2-x²) / x² and y = cos(2sec^-1x)). In addition to comparing the functions graphically, show analytically why your answer holds.

Problem 2

Suppose f(x) is a function whose general antiderivative is F(x) + C, i.e.,

Now let h(x) = f(x+k) for any constant k. Show that

(Lots of integrations by partial fractions use this rule implicitly, so it’s nice to know that it’s true in general.)

Problem 3

Exercise 2 in section 8.4 of the textbook (integrate (3dx / √(1+9x²)).

Problem 4

Exercise 53 in section 8.4 of the textbook (find the area between the curve y = √(9-x²) / 3 and the axes)).

Problem 5

Exercise 10 in section 8.5 of the textbook (integrate 1 / (x²+2x)).

Problem 6

Exercise 20 in section 8.5 of the textbook (integrate x² / ( (x-1)(x²+2x+1) )).

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.