Day
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4.5 Integration by u-Substitution
Substitution in Definite Integrals
two routes:
1. transform the integral as an indefinite integral, integrate, change back to the original variable, and use the original limits of integration
2. make the same u­substitution you would use to evaluate the corresponding indefinite integral, transform the limits of integration, then evaluate using the transformed limits
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Examples:
Evaluate.
/4
tan x . sec 2x dx
1. 0
use both routes
verify using fnInt
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6
3
x2. x + 2
dx
2. [ p298 #78 ]
­2
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Further Examples:
Slope Fields
[ Exercise 40 (page 297) ]
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[ Exercise #90 (page 298) ]
Find the indefinite integral in two ways.
Explain any difference in the forms of the answers.
sin x . cos x dx
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[ Exercise #84 (page 298) ]
2
3
x
x + 2
dx
0
. . . use a graphing utility to evaluate the integral. Graph the region whose area is given by the definite integral.
2nd CALC
7. f(x) dx
[­1, 3]
[­5, 20]
lower limit
x = 0 enter
upper limit
x = 2 enter
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Assignment
p297 #35, 37, 39, 55
p298 #65-85 odd,
#103 Application - Depreciation Rates,
#105 Application - average sales (yav) see Example 5 p280
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