Questions and Answers #41 Applications of Integrals - Edubirdie

University:
Stanford University
Course:
MATH 52 | Integral Calculus of Several Variables
Academic year:

2021
Views:

6

Pages:

1
Author:

MathPaladin

Questions and Answers Sheet 41

Applications of Integrals Question #1 𝑛π Why does ∫0 𝑑𝑥 1+tan2𝑘 (𝑥) =𝑛 π 2 hold for all non-negative integer k? Answer: 𝑛π ∫0 𝑑𝑥 1+tan2𝑘 𝑥 The integrand is periodic with period π and symmetric within a period about π/2 = 2𝑛 ∫0 𝑑𝑥 the two integrands is 1: 2𝑛 2 π/2 ∫0 2 1+tan2𝑘 𝑥 Swap the integral bounds. The integrand then evaluates to = π 1 𝑑𝑥 = 𝑛π 2 tan2𝑘 1+tan2𝑘 𝑥 by tan(π/2 − 𝑥) = cot 𝑥. The sum of

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