1. Integration by Substitution
4.5 Day 1
On the agenda:
How to Integrate using u-substitution aka Change of Variable(s)
HW: p. 297 # 3-33 mx3, odd

2. Role of Substitution in Integration
Substitution is a technique that we can sometimes use to rewrite functions in a form that we can integrate.
Recognition is KEY!
Two Components to keep in mind:
Look for a derivative one term to be somewhere in the integral.
Do not worry about coefficients since we can ‘adjust’ for them

3. Don’t forget to substitute the value for u back into the problem!
Example 1
Don’t forget to substitute the value for u back into the problem!
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4. The variable of integration must match the variable in the expression.
Example 2:
The variable of integration must match the variable in the expression.
Don’t forget to substitute the value for u back into the problem!
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5. Note that this only worked because of the 2x in the original.
Example 3:
The derivative of is
Note that this only worked because of the 2x in the original.
Many integrals can not be done by substitution.
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6. You Try It First
Example 4:
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7. Here U-Sub does not work
Example 5
Here U-Sub does not work
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8. You Try It First
Example 6:
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9. You Try It First
Example 7:
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10. You Try It First
Example 8:
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12. Change of Variables Wrong! The limits don’t match! Example 5:
Using the original limits:
Leave the limits out until you substitute back.
Wrong!
The limits don’t match!
This is usually more work than finding new limits
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13. We can find new limits, and then we don’t have to substitute back.
Example 5:
new limit
We can find new limits, and then we don’t have to substitute back.
new limit
We could have substituted back and used the original limits.
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14. Don’t forget to use the new limits.
Example 6:
Don’t forget to use the new limits.
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