1. Section 8.2 Calculus BC AP/Dual, Revised ©2019 viet.dang@humbleisd.net
Integration by Parts
Section 8.2
Calculus BC AP/Dual, Revised ©2019
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§8.2: Integration by Parts

2. §8.2: Integration by Parts
Review
Product Rule: Integration of Product Rule
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§8.2: Integration by Parts

3. §8.2: Integration by Parts
Integrate by Parts
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§8.2: Integration by Parts

4. §8.2: Integration by Parts
Used to simplify integrals of the form, 𝒇 𝒙 𝒈 𝒙 𝒅𝒙
These products include algebraic and transcendental functions, such as inverse, logarithmic, trigonometric, and exponential functions
Used to assist in taking the integral of a complicated problem
Equation: 𝒖𝒅𝒗 =𝒖𝒗− 𝒗𝒅𝒖 “ultra-violet voo-doo”
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§8.2: Integration by Parts

5. §8.2: Integration by Parts
Steps
Determine the appropriate substitution for 𝒖 and 𝒅𝒗 where 𝒖 is the most simple and 𝒅𝒗 is the most complicated form
To Establish 𝒖, “𝑳𝑰𝑨𝑻𝑬 for 𝑼” is BASED ON ORDER TO DETERMINE WHAT IS “𝑼”
Logarithmic
Inverse trigonometric
Algebraic
Trigonometric
Exponential
Integrate 𝒅𝒗 to solve for 𝒗, and differentiate 𝒖 to solve for 𝒅𝒖.
Substitute the values for 𝒖, 𝒗, 𝒅𝒖, and 𝒅𝒗 into the formula and solve
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§8.2: Integration by Parts

6. §8.2: Integration by Parts
LIATE Examples
To Establish 𝒖, “𝑳𝑰𝑨𝑻𝑬 for 𝑼” is BASED ON ORDER TO DETERMINE WHAT IS “𝑼”
Logarithmic
𝐥𝐧 𝒙 , 𝐥𝐨𝐠 𝟐 𝒙 ,𝒆𝒕𝒄…
Inverse trigonometric
𝐭𝐚𝐧 −𝟏 𝒙 , 𝐚𝐫𝐜𝐬𝐢𝐧 𝒙 ,𝒆𝒕𝒄…
Algebraic
𝒙, 𝒙 𝟐 ,𝒙+𝟏,𝒆𝒕𝒄…
Trigonometric
𝐜𝐨𝐬 𝒙 ,𝐭𝐚𝐧(𝒙),𝒆𝒕𝒄…
Exponential
𝒆 𝒙 , 𝟐 𝒙 ,𝒆𝒕𝒄…
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§8.2: Integration by Parts

7. §8.2: Integration by Parts
Example 1
Solve 𝒙 𝒆 𝟐𝒙 𝒅𝒙
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§8.2: Integration by Parts

8. §8.2: Integration by Parts
Example 1
Solve 𝒙 𝒆 𝟐𝒙 𝒅𝒙
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§8.2: Integration by Parts

9. §8.2: Integration by Parts
Example 2
Solve 𝒙 𝐥𝐧𝒙 𝒅𝒙
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§8.2: Integration by Parts

10. §8.2: Integration by Parts
Example 2
Solve 𝒙 𝐥𝐧𝒙 𝒅𝒙
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§8.2: Integration by Parts

11. §8.2: Integration by Parts
Your Turn
Solve 𝒙 𝟒 𝐥𝐧𝒙 𝒅𝒙
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§8.2: Integration by Parts

12. §8.2: Integration by Parts
Example 3
Solve 𝟒 𝐚𝐫𝐜𝐜𝐨𝐬𝒙 𝒅𝒙
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§8.2: Integration by Parts

13. §8.2: Integration by Parts
Example 3
Solve 𝟒 𝐚𝐫𝐜𝐜𝐨𝐬𝒙 𝒅𝒙
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§8.2: Integration by Parts

14. §8.2: Integration by Parts
Example 4
Solve 𝒆 𝒙 𝐜𝐨𝐬𝒙 𝒅𝒙
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§8.2: Integration by Parts

15. §8.2: Integration by Parts
Your Turn
Solve 𝒙 𝟐 𝐜𝐨𝐬𝒙 𝒅𝒙
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§8.2: Integration by Parts

16. §8.2: Integration by Parts
Example 5
Solve 𝐥𝐧 𝒙 𝒅𝒙
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§8.2: Integration by Parts

17. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝒙 𝟐 𝐜𝐨𝐬 𝒙 𝒅𝒙=𝒇 𝒙 − 𝟐𝒙 𝐬𝐢𝐧 𝒙 𝒅𝒙, then 𝒇 𝒙 = (A) 𝟐 𝐬𝐢𝐧 𝒙 + 𝟐𝒙 𝐜𝐨𝐬𝒙+𝑪 (B) 𝒙 𝟐 𝐬𝐢𝐧 𝒙+𝑪 (C) 𝟐𝒙 𝐜𝐨𝐬 𝒙− 𝒙 𝟐 𝐬𝐢𝐧 𝒙+𝑪 (D) 𝟐− 𝒙 𝟐 𝐜𝐨𝐬 𝒙−𝟒 𝐬𝐢𝐧 𝒙+𝑪
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§8.2: Integration by Parts

18. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝒙 𝟐 𝐜𝐨𝐬 𝒙 𝒅𝒙=𝒇 𝒙 − 𝟐𝒙 𝐬𝐢𝐧 𝒙 𝒅𝒙, then 𝒇 𝒙 =
Vocabulary
Connections and Process
Answer
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§8.2: Integration by Parts

19. §8.2: Integration by Parts
Assignment
Page odd, odd, 31, 32
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§8.2: Integration by Parts