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Peer Questions for Section 6.1
[Submit your responses by 3 pm, Wed., Sept. 4, using the webform below.]
1. Once you had learned the material of Chapter 5 of our text (i.e., Fundamental Theorem of
Calculus, etc.), you began writing statements like these:
ż
ż
ż
5 3
dx
2
5x dx “ x ` C,
sin x dx “ ´ cos x ` C,
“ arctan x ` C,
etc.
3
1 ` x2
What do statements such as these actually say? [Hint: Use the term antiderivative in your
response.]
ş
2. When we write cos x dx “ sin x ` C, the content of this mathematical statement can be
phrased in terms of antiderivatives (as in Question 1). But it can also be phrased in terms
of derivatives: ”Functions of the form sin x ` C have, as their derivative, the function cos x.”
When viewed this way, what familiar rule for differentiation does the integration by parts
formula reiterate?
[Note: Be able to state the integration by parts formula, as it is stated in Box 1 or Box 2 on
p. 312, at any time.]
3. Name some simple-looking functions for which you have not learned (at least, not from
material found in Chapters 1–5 of our text), the precise form of their antiderivatives.
4. Integration by parts is a systematic method for finding antiderivatives of some of the functions that fit the criterion of Question 3 (i.e., ones whose antiderivatives were not presented
in Chapters 1–5). Write down at least 3 examples of simple-looking functions whose antiderivatives are not obvious, but may be found using integration by parts. Among the 3
examples, choose one that requires two iterations of integration by parts before the answer is
known. Submit these functions as your answer to Question 4.
Though no further answers to Question 4 should be submitted online, take each of the
three functions f you wrote and use integration by parts (more than once, when necessary)
to find the form of antiderivatives of f . You may check your work using the website
http://www.wolframalpha.com.
5. Identify one item (a concept, a step in an example, a statement, etc.) from this reading
assignment you found difficult or confusing.