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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 8 - Integration Techniques문제 8.R.118b
8장, 문제 8.R.118b

118. Two worthy integrals
b. Let f be any positive continuous function on the interval [0, π/2]. Evaluate
∫ from 0 to π/2 of [f(cos x) / (f(cos x) + f(sin x))] dx.
(Hint: Use the identity cos(π/2 − x) = sin x.)


(Source: Mathematics Magazine 81, 2, Apr 2008)

검증된 단계별 안내
1
Step 1: Recognize the integral and the hint provided. The integral is ∫ from 0 to π/2 of [f(cos x) / (f(cos x) + f(sin x))] dx, and the hint suggests using the trigonometric identity cos(π/2 − x) = sin x.
Step 2: Apply the substitution u = π/2 − x. This substitution changes the limits of integration: when x = 0, u = π/2; and when x = π/2, u = 0. Also, dx becomes -du.
Step 3: Rewrite the integral using the substitution. The integral becomes ∫ from π/2 to 0 of [f(cos(π/2 − u)) / (f(cos(π/2 − u)) + f(sin(π/2 − u)))] (-du). Using the identity cos(π/2 − u) = sin u and sin(π/2 − u) = cos u, simplify the integrand.
Step 4: Simplify the integral further. After substitution, the integral becomes ∫ from π/2 to 0 of [f(sin u) / (f(sin u) + f(cos u))] (-du). Reverse the limits of integration to remove the negative sign, resulting in ∫ from 0 to π/2 of [f(sin x) / (f(sin x) + f(cos x))] dx.
Step 5: Observe symmetry in the integrand. The original integral and the transformed integral are symmetric, meaning their values are equal. Combine the two integrals to conclude that the value of the integral is 1/2, based on symmetry and equal contributions from both parts.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definite Integrals

A definite integral represents the signed area under a curve defined by a function over a specific interval. In this case, the integral from 0 to π/2 of the function f(cos x) / (f(cos x) + f(sin x)) is evaluated, which requires understanding how to compute areas and the properties of integrals.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral

Trigonometric Identities

Trigonometric identities, such as cos(π/2 - x) = sin x, are fundamental relationships between trigonometric functions. This identity is crucial for transforming the integral and simplifying the evaluation process, allowing for a substitution that can lead to a more manageable expression.
추천 영상:
7:17
Verifying Trig Equations as Identities

Symmetry in Integrals

Symmetry in integrals refers to the property that certain functions exhibit when evaluated over symmetric intervals. In this problem, recognizing that the integral can be manipulated using the identity to show that the integral of f(sin x) has a similar structure to that of f(cos x) helps in simplifying the evaluation and finding a solution.
추천 영상:
가이드 코스
06:18
Integration by Parts for Definite Integrals
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