1. Section 4.5A Calculus AP/Dual, Revised ©2019 viet.dang@humbleisd.net
Change of Variables
Section 4.5A
Calculus AP/Dual, Revised ©2019
2/18/2019 6:04 PM
§4.5A: Change of Variables

2. §4.5A: Change of Variables
Use a Change of Variables to evaluate a definite integral
When applying 𝒖-substitution to evaluate a definite integral, make sure to change the Limits of Integration before evaluating the integral
If the function 𝒖=𝒈(𝒙) has a continuous derivative on the closed interval 𝒂, 𝒃 and f is continuous on the range of g, then: 𝒂 𝒃 𝒇(𝒈(𝒙))𝒈′(𝒙)𝒅𝒙= 𝒈 𝒂 𝒈 𝒃 𝑭(𝒈(𝒙))+𝑪
𝒂 = Lower Bound, 𝒃 = Upper Bound
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§4.5A: Change of Variables

3. §4.5A: Change of Variables
Steps
Simplify/Rewrite the problem
Establish the 𝒖 as the base, 𝒅𝒖 as the derivative, and 𝒅𝒙 as the constant outside
Integrate: “Add and Divide”
Simplify
Apply 𝒃 and 𝒂 into the 𝒖-substitution equation because of the new limits and subtract
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§4.5A: Change of Variables

4. §4.5A: Change of Variables
Example 1
Solve 𝟎 𝟏 𝒙 𝒙 𝟐 +𝟏 𝟑 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

5. §4.5A: Change of Variables
Example 1
Solve 𝟎 𝟏 𝒙 𝒙 𝟐 +𝟏 𝟑 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

6. §4.5A: Change of Variables
Example 2
Solve 𝟎 𝟏 𝒙 𝟐 𝒙 𝟑 +𝟖 𝟐 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

7. §4.5A: Change of Variables
Your Turn
Solve 𝟏 𝟓 𝟐𝒙−𝟏 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

8. §4.5A: Change of Variables
Example 3
Solve 𝟏 𝟓 𝒙 𝟐𝒙−𝟏 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

9. §4.5A: Change of Variables
Example 3
Solve 𝟏 𝟓 𝒙 𝟐𝒙−𝟏 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

10. §4.5A: Change of Variables
Example 3
Solve 𝟏 𝟓 𝒙 𝟐𝒙−𝟏 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

11. §4.5A: Change of Variables
Example 4
Solve 𝟎 𝟒 𝟏 𝟐𝒙+𝟏 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

12. §4.5A: Change of Variables
Your Turn
Solve 𝟎 𝟐 𝒙 𝟏+𝟐 𝒙 𝟐 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

13. §4.5A: Change of Variables
Example 5
Solve −𝝅 𝝅 𝟐 𝐜𝐨𝐬 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

14. §4.5A: Change of Variables
Example 6
Solve 𝟎 𝝅/𝟐 𝟒 𝐬𝐢𝐧 𝒙 𝟐 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

15. §4.5A: Change of Variables
Your Turn
Solve 𝟎 𝝅/𝟒 𝟒 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

16. §4.5A: Change of Variables
Example 7
Solve 𝟎 𝟖 𝒆 −𝟒𝒙 𝒅𝒙 using change of variables
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§4.5A: Change of Variables

17. §4.5A: Change of Variables
Example 8
Solve 𝟎 𝟏 𝟐 𝟏+ 𝒙 𝟐 𝒅𝒙 using change of variables
2/18/2019 6:04 PM
§4.5A: Change of Variables

18. AP Multiple Choice Question 1 (non-calculator)
Using the substitution, 𝒖= 𝒙 , 𝟏 𝟒 𝒆 𝒙 𝒙 𝒅𝒙 is equal to the following?
𝟐 𝟏 𝟒 𝒆 𝒖 𝒅𝒖
𝟐 𝟏 𝟐 𝒆 𝒖 𝒅𝒖
𝟏 𝟐 𝟏 𝟐 𝒆 𝒖 𝒅𝒖
𝟏 𝟒 𝒆 𝒖 𝒅𝒖
2/18/2019 6:04 PM
§4.5A: Change of Variables

19. AP Multiple Choice Question 1 (non-calculator)
Using the substitution, 𝒖= 𝒙 , 𝟏 𝟒 𝒆 𝒙 𝒙 𝒅𝒙 is equal to the following?
Vocabulary
Connections and Process
Answer and Justifications
2/18/2019 6:04 PM
§4.5A: Change of Variables

20. §4.5A: Change of Variables
Assignment
Worksheet
2/18/2019 6:04 PM
§4.5A: Change of Variables