Questions and Answers #1 Antiderivatives - Edubirdie

University:
Stanford University
Course:
MATH 115 | Functions of a Real Variable
Academic year:

2021
Views:

5

Pages:

1
Author:

MathPaladin

Questions and Answers Sheet 1 Antiderivatives Question #1 Let F(x) be an antiderivative of (2(𝑙𝑛𝑥)4 )/(3𝑥) , If F(2)=0, then F(8)=? Answer: We have: 4 𝑓(𝑥) = (2(𝑙𝑛(𝑥)) ) /(3𝑥) We need integrate. Let tau equiv ln(x) . 𝐹(𝑥) = ∫ (2𝜏 4 )/(3)𝑑 𝜏 𝐹(𝑥) = 2/15𝜏 5 + 𝐶 5 𝐹(𝑥) = 2/15(𝑙𝑛(𝑥)) + 𝐶 where C is a randomly chosen integration constant. We have more information, that F(2) = 0 : 5 𝐹(2) = 2/15(𝑙𝑛(2)) + 𝐶 = 0 5 𝐶 = −2/15(𝑙𝑛(2)) Then, we find: 5 5 5 5 𝐹(𝑥) = 2/15(𝑙𝑛(𝑥)) − 2/15(𝑙𝑛(2)) If we substitute x = 8 now: 𝐹(8) = 2/15(𝑙𝑛(8)) − 2/15(𝑙𝑛(2)) ≈ 5.163

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