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)

Verified step by step guidance

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.
2. 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.
3. 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.
4. 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.
5. 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.

Key Concepts

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.

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.

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.