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Exercise 3
Math Animated™
Mathematical Introduction for Physics and Engineering
by Samuel Dagan (Copyright © 2007-2020)
Chapter 2: Integration; Section 1: Indefinite Integrals; page 3
Special Methods, Exercise 3
Question
Use the substitution in order to calculate the indefinite integral:
Reminder
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(2.1.1.18)
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One very general method for converting an integral to a familiar form is called integration by substitution ...
:
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(2.1.2.1)
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(2.1.2.4)
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... trigonometric substitution ...
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(2.1.3.11)
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from which one obtains the differential relation
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(2.1.3.12)
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Parts
- By use of (2.1.3.11) one obtains
- which together with (2.1.3.12) yields
- The integral can be solved by the use of the following substitutions (2.1.2.4)
- The integral becomes
- The substitution back of
gives the requirement that |v|<1 or
corresponds to
- From part 5, a numerical calculation yields
or
- Finally we get from parts 5 and 6
Score
Parts 1,2,3,6 are worth 1 point each.
Parts 4,5,7 are worth 2 points each.