Ch. 4.8, Rules of Antiderivatives | Georgia Southern University

University:
Georgia Southern University
Course:
MATH 1441 | Calculus I
Academic year:

2024
Views:

463

Pages:

6
Author:

Jalen W.

Antidifferentiation: Rules of Antidifferentiation Integration / Antiderivatives : A function 𝑭 is an antiderivative of 𝒇 on an interval if 𝑭′ (𝒙) = 𝒇(𝒙) 𝑖. 𝑒. ∫ 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑥) 𝑖𝑓 𝑭′ (𝒙) = 𝒇(𝒙) To find an antiderivative of 𝑓(𝑥) = 3𝑥 2 : Since Since 𝑑 𝑑𝑥 𝑑 𝑑𝑥 (𝑥 3 ) = 3𝑥 2 , 𝐹(𝑥) = 𝑥 3 is an antiderivative of 𝑓(𝑥). ∫ 3𝑥 2 𝑑𝑥 = 𝑥 3 (𝑥 3 − 5) = 3𝑥 2 , 𝐹(𝑥) = (𝑥 3 − 5) is an antiderivative of 𝑓(𝑥). ∫ 3𝑥 2 𝑑𝑥 = 𝑥 3 − 5 Since 𝑑 𝑑𝑥 (𝑥 3 + 97) = 3𝑥 2 , 𝐹(𝑥) = (𝑥 3 + 97) is an antiderivative of 𝑓(𝑥). Example: Verify that 𝐹 is an antiderivative of 𝑓. 1. 𝐹(𝑥) = 𝑥𝑙𝑛 𝑥 − 𝑥; if 𝑭′ (𝒙) = 𝒇(𝒙). 𝑓(𝑥) = ln 𝑥, or ∫ ln 𝑥 𝑑𝑥 = 𝑥𝑙𝑛 𝑥 − 𝑥 1 2. 𝐹(𝑥) = 4 ln 𝑥 + 3𝑥 + 𝑥 4 + 10, 2 ∫ 4+3𝑥+2𝑥 4 𝑓(𝑥) = 4+3𝑥+2𝑥 4 𝑥 1 𝑑𝑥 = 4 ln 𝑥 + 3𝑥 + 𝑥 4 + 10 if 𝑭′ (𝒙) = 𝒇(𝒙) 𝑥 2 Basic Integration Rules: Differentiation Formula 1. 2. 3. 4. 5. 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 Integration Formula [𝐶] = 0 ∫ 0 𝑑𝑥 = 𝐶 [𝑘𝑥] = 𝑘 ∫ 𝑘 𝑑𝑥 = 𝑘𝑥 + 𝐶 ∫ 𝑥 𝑛 𝑑𝑥 = [𝑘 𝑓(𝑥)] = 𝑘 𝑓 ′ (𝑥) ∫ 𝑘 𝑓(𝑥) 𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥)𝑑𝑥 [𝑓(𝑥) ± 𝑔(𝑥)] 7. 𝑑 𝑑𝑥 𝑑 𝑑𝑥 [𝑒 𝑥 ] = 𝑒 𝑥 [ln|𝑥|] = 1 𝑥 𝑛+1 + 𝐶, 𝑖𝑓 𝑛 ≠ −1 ∫[𝑓(𝑥) ± 𝑔(𝑥)] 𝑑𝑥 = 𝑓 ′ (𝑥 ) ± 𝑔′ (𝑥 ) 6. 𝑥 𝑛+1 [𝑥 𝑛 ] = 𝑛𝑥 𝑛−1 = ∫ 𝑓(𝑥 )𝑑𝑥 ± ∫ 𝑔(𝑥 )𝑑𝑥 ∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶 1 ∫ 𝑥 −1 𝑑𝑥 = ∫ 𝑥 𝑑𝑥 = ln|𝑥| + 𝐶 Product rule of differentiation Quotient Rule of differentiation Examples: Find the indefinite integral and check the result by differentiation. 1. ∫ 𝑥 7 𝑑𝑥 1 ∫ 𝑥 3 𝑑𝑥 ∫ √𝑥 𝑑𝑥; 2. ∫(3𝑥 5 + 7𝑥 2 + 8)𝑑𝑥 3. ∫(3𝑥 + 2)2 𝑑𝑥 𝑥 4 −1 4. ∫ ( 5. 𝑥2 ) 𝑑𝑥 ∫ 𝑡 3 (𝑡 2 + 𝑡 − 1)𝑑𝑡