|
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Line 5: |
Line 5: |
| | | |
| {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" |
| + | == 1 == |
| !Foundations: | | !Foundations: |
| |- | | |- |
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Line 31: |
| | | |
| '''Solution:''' | | '''Solution:''' |
− | | + | == 2 == |
| '''(a)''' | | '''(a)''' |
| | | |
Line 94: |
Line 95: |
| | | | | |
| |} | | |} |
− | | + | == 3 == |
| '''(b)''' | | '''(b)''' |
| | | |
Line 128: |
Line 129: |
| \end{array}</math> | | \end{array}</math> |
| |} | | |} |
− | | + | == 4 == |
| {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" |
| !Final Answer: | | !Final Answer: |
Revision as of 22:57, 25 February 2016
Evaluate the improper integrals:
- a)

- b)

1
ExpandFoundations:
|
1. How could you write so that you can integrate?
|
- You can write

|
2. How could you write ?
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- The problem is that
is not continuous at .
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- So, you can write
.
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3. How would you integrate ?
|
- You can use integration by parts.
|
- Let
and .
|
Solution:
2
(a)
ExpandStep 1:
|
First, we write .
|
Now, we proceed using integration by parts. Let and . Then, and .
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Thus, the integral becomes
|

|
ExpandStep 2:
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For the remaining integral, we need to use -substitution. Let . Then, .
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Since the integral is a definite integral, we need to change the bounds of integration.
|
Plugging in our values into the equation , we get and .
|
Thus, the integral becomes
|

|
ExpandStep 3:
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Now, we evaluate to get
|

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Using L'Hopital's Rule, we get
|

|
|
3
(b)
ExpandStep 1:
|
First, we write .
|
Now, we proceed by -substitution. We let . Then, .
|
Since the integral is a definite integral, we need to change the bounds of integration.
|
Plugging in our values into the equation , we get and .
|
Thus, the integral becomes
|
.
|
ExpandStep 2:
|
We integrate to get
|

|
4
ExpandFinal Answer:
|
(a)
|
(b)
|
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