Question

Evaluate the definite integral using standard definite integral properties: \[ \int_{0}^{\frac{\pi}{2}} \frac{\sin^5 x}{\sin^5 x + \cos^5 x} \, dx \]

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{2} \)
• C. \( \pi \)
• D. \( 0 \)

Hint:

For any definite integral problem bounded precisely from \( 0 \) to \( \frac{\pi}{2} \) that follows the standard structural form \( \frac{f(\sin x)}{f(\sin x) + f(\cos x)} \), the answer will always evaluate to exactly half of the upper bound: \( \frac{\pi}{4} \).

Answer 1

The Correct Option is A

Concept: According to the King's Property (\( P_4 \)) of definite integrals, substituting the sum of the boundary limits minus the independent variable leaves the net value of the definite integral completely unchanged: \[ I = \int_{a}^{b} f(x)\,dx = \int_{a}^{b} f(a+b-x)\,dx \]

Step 1: Apply the boundary property to create an alternative equation.
Let our initial integral equation be designated as equation (1): \[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^5 x}{\sin^5 x + \cos^5 x} \, dx \quad \cdots (1) \] Apply the definite boundary property by replacing every \( x \) variable with \( \left(0 + \frac{\pi}{2} - x\right) = \frac{\pi}{2} - x \): \[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^5(\frac{\pi}{2} - x)}{\sin^5(\frac{\pi}{2} - x) + \cos^5(\frac{\pi}{2} - x)} \, dx \] Using standard co-function reduction identities where \( \sin(\frac{\pi}{2} - x) = \cos x \) and \( \cos(\frac{\pi}{2} - x) = \sin x \), rewrite the equation: \[ I = \int_{0}^{\frac{\pi}{2}} \frac{\cos^5 x}{\cos^5 x + \sin^5 x} \, dx \quad \cdots (2) \]

Step 2: Add both equations together to simplify the integrand expression.
Add equation (1) and equation (2) together. Since they share identical integration boundaries and denominators, combine their numerators directly: \[ 2I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^5 x + \cos^5 x}{\sin^5 x + \cos^5 x} \, dx \] The matching numerator and denominator terms cancel out completely, leaving an integrand value of 1: \[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \, dx = [x]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2} \]

Step 3: Isolate the single integral variable \( I \).
Divide both sides of the equation by 2 to find the final value: \[ 2I = \frac{\pi}{2} \quad \Rightarrow \quad I = \frac{\pi}{4} \]