118. Two worthy integrals
b. Let f be any positive conti...

Question

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)

Accepted Answer

Let H be any positive continuous function on the interval from 0 to pi divided by 2. Compute the interval from 0 to pi divided by 2 of H tangent X divided by H tangent X plus H cotangent XD X. There are 4 possible inserts being pi, pi divided by 4, pi divided by 2, and pi divided by 3. Now, to solve this, we'll first make use of a couple of properties. First, You know that We can write our integral from 0 to a. Of FXDX. As the integral from 0 to A F of A minus XD X. If F is integrable on the interval from 0 to 8. We also know that the tangent of pi divided by 2 minus X equals cotangent. And vice versa. cotangent of pi divided by 2 minus X equals tangent. So now what we're going to do is rewrite our integral. We have H of. Tangent I divided by 2 minus X. Defined by H of tangent, I divided by 2 minus X. Plus H cotangent. I divided by 2 minus X. Which gives us H. Cotangent X. Divided by H of cotangent X. Plus H of tangent X. We cannot write our new integral as the integral from 0 to pi divided by 2. Of H Cotangent X divided by H cotangent X plus H tangent X. With DX at the end. Now, what we'll do is add our original integral and substitute the integrals together. We have the integral from 0 to the pi divided by 2 of H. Of tangent X. Divided by H X plus H coats X. Plus H of cotangent X. Divided by H of cotangent X plus H tangent X. TX. Now, we end up getting 2 i equals the integral from 0, the pi divided by 2. Of H tangent X plus H. Code tangent X. Divided by H tangent X plus H cotangent X. Since they have the same denominator, we can combine them. Meaning we now have 2 i equals the integral from 0, the pi divided by 2 of 1 DX. This is given by pi divided by 2. We now have 2 i equals pi divided by 2. And if we divide both sides by 2, we get i equals pi divided by 4. This is a result of our integral, which means the answer to our problem is answer B. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

118. Two worthy integralsb. 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)

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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)