1. Antiderivatives & Indefinite Integration
Section 4.1
Calculus AP/Dual, Revised ©2018
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§4.1: Antiderivatives

2. Brief History Gottfried Wilhelm Leibniz vs Sir Issac Newton
Newton used Derivatives to solve equations whereas Leibniz used Integration  
Leibniz, “saw as a generalization of the summation of infinite series, whereas Newton began from derivatives” (Wikipedia)
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§4.1: Antiderivatives

3. Definitions
Written as 𝑭 (known as the anti), is an antiderivative of 𝒇 in the interval of 𝒇 ′ 𝒙 = 𝒇 𝒙 for all 𝒙 in 𝑰.
It is also known as the “original function” and “general solutions” due to its constant of integration
The “+𝑪 ” stands for the Constant of Integration (or constant)
The symbol means “infinite sum”
Indefinite Integral has no limits
Definite Integral has a starting and ending limit
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§4.1: Antiderivatives

4. Breaking Down the Integral
Constant
Integral
Variable of Integration
Integrand
“The antiderivative of 𝒇 with respect to 𝒙"
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§4.1: Antiderivatives

5. Guess The Rule 𝒙 𝟐 +𝑪 𝒙 𝟑 +𝑪 𝒙 𝟒 +𝑪 𝒙 𝟐 +𝑪 𝒙 𝟐 +𝑪 𝒙 𝟑 +𝑪 𝒙 𝟒 +𝑪 𝒙 𝟐 +𝑪
𝟐𝒙𝒅𝒙
𝒙 𝟐 +𝑪
𝟑𝒙 𝟐 𝒅𝒙
𝒙 𝟑 +𝑪
𝟒𝒙 𝟑 𝒅𝒙
𝒙 𝟒 +𝑪
𝒙 𝟒 𝒅𝒙
𝟑𝒙 𝟓 𝒅𝒙
𝒙 𝒏 𝒅𝒙
𝟐𝒙𝒅𝒙
𝟑𝒙 𝟐 𝒅𝒙
𝟒𝒙 𝟑 𝒅𝒙
𝒙 𝟒 𝒅𝒙
𝟑𝒙 𝟓 𝒅𝒙
𝒙 𝒏 𝒅𝒙
𝟐𝒙𝒅𝒙
𝒙 𝟐 +𝑪
𝟑𝒙 𝟐 𝒅𝒙
𝟒𝒙 𝟑 𝒅𝒙
𝒙 𝟒 𝒅𝒙
𝟑𝒙 𝟓 𝒅𝒙
𝒙 𝒏 𝒅𝒙
𝟐𝒙𝒅𝒙
𝒙 𝟐 +𝑪
𝟑𝒙 𝟐 𝒅𝒙
𝒙 𝟑 +𝑪
𝟒𝒙 𝟑 𝒅𝒙
𝒙 𝟒 +𝑪
𝒙 𝟒 𝒅𝒙
𝟏 𝟓 𝒙 𝟓 +𝑪
𝟑𝒙 𝟓 𝒅𝒙
𝒙 𝒏 𝒅𝒙
𝟐𝒙𝒅𝒙
𝒙 𝟐 +𝑪
𝟑𝒙 𝟐 𝒅𝒙
𝒙 𝟑 +𝑪
𝟒𝒙 𝟑 𝒅𝒙
𝒙 𝟒 +𝑪
𝒙 𝟒 𝒅𝒙
𝟏 𝟓 𝒙 𝟓 +𝑪
𝟑𝒙 𝟓 𝒅𝒙
𝟏 𝟐 𝒙 𝟔 +𝑪
𝒙 𝒏 𝒅𝒙
𝒙 𝒏+𝟏 𝒏+𝟏 +𝑪
𝟐𝒙𝒅𝒙
𝒙 𝟐 +𝑪
𝟑𝒙 𝟐 𝒅𝒙
𝒙 𝟑 +𝑪
𝟒𝒙 𝟑 𝒅𝒙
𝒙 𝟒 𝒅𝒙
𝟑𝒙 𝟓 𝒅𝒙
𝒙 𝒏 𝒅𝒙
𝟐𝒙𝒅𝒙
𝒙 𝟐 +𝑪
𝟑𝒙 𝟐 𝒅𝒙
𝒙 𝟑 +𝑪
𝟒𝒙 𝟑 𝒅𝒙
𝒙 𝟒 +𝑪
𝒙 𝟒 𝒅𝒙
𝟏 𝟓 𝒙 𝟓 +𝑪
𝟑𝒙 𝟓 𝒅𝒙
𝟏 𝟐 𝒙 𝟔 +𝑪
𝒙 𝒏 𝒅𝒙
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§4.1: Antiderivatives

6. Steps Simplify/Rewrite the problem
Integrate: “Add the exponent, Divide the new exponent, and Conquer the constant”
Simplify
Rule: (Power Rule in Reverse) 𝒙 𝒏 𝒅𝒙= 𝒙 𝒏+𝟏 𝒏+𝟏 +𝑪
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§4.1: Antiderivatives

7. Basic Integration Rules
𝟎𝒅𝒙=𝑪
𝒌𝒅𝒙=𝒌𝒙+𝑪
𝒌𝒇 𝒙 𝒅𝒙=𝒌 𝒇 𝒙 𝒅𝒙=𝒌𝒇 𝒙 +𝑪
𝒇 𝒙 ±𝒈 𝒙 𝒅𝒙= 𝒇 𝒙 𝒅𝒙± 𝒈 𝒙 𝒅𝒙= 𝒇 𝒙 ±𝒈 𝒙 +𝑪
𝒙 𝒏 𝒅𝒙= 𝒙 𝒏+𝟏 𝒏+𝟏 +𝑪 (Add, Divide, and Conquer)
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§4.1: Antiderivatives

8. Review Solve 𝒅𝒚 𝒅𝒙 for 𝒇(𝒙)=𝒙 𝟑 1/18/2019 2:06 PM
§4.1: Antiderivatives

9. Example 1
Solve 𝟑𝒙 𝟐 𝒅𝒙
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§4.1: Antiderivatives

10. Example 2
Solve 𝒙 𝟑 −𝟒𝒙+𝟐 𝒅𝒙
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§4.1: Antiderivatives

11. Example 3 Solve 𝟖𝒙 𝟑 + 𝟏 𝟐 𝒙 𝟐 𝒅𝒙 1/18/2019 2:06 PM
§4.1: Antiderivatives

12. Your Turn Solve 𝒙 𝟑 𝟒 + 𝟏 𝒙 𝟐 𝒅𝒙 1/18/2019 2:06 PM
§4.1: Antiderivatives

13. Example 4 Solve 𝒙 𝟐 +𝟐𝒙−𝟑 𝒙 𝟏/𝟐 𝒅𝒙 1/18/2019 2:06 PM
§4.1: Antiderivatives

14. Example 5
Solve 𝟐 𝜽 𝟐 −𝟑𝜽+𝟏 𝒅𝜽
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§4.1: Antiderivatives

15. Your Turn
Solve 𝟐𝒕+𝟑 𝟐 𝒅𝒕
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§4.1: Antiderivatives

16. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝒙 𝟑 −𝟑𝒙 𝒅𝒙 (A) 𝟑 𝒙 𝟐 −𝟑+𝑪 (B) 𝟒 𝒙 𝟒 −𝟔 𝒙 𝟐 +𝑪 (C) 𝒙 𝟒 𝟒 −𝟑𝒙+𝑪 (D) 𝒙 𝟒 𝟒 − 𝟑𝒙 𝟐 𝟐 +𝑪
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§4.1: Antiderivatives

17. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝒙 𝟑 −𝟑𝒙 𝒅𝒙
Vocabulary
Process and Connections
Answer and Justifications
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§4.1: Antiderivatives

18. Assignment
Page 251
7-24 all (No need to check answers as stated in questions 11-24)
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§4.1: Antiderivatives