Step 1: Understanding the Question:
We need to evaluate the definite integral \(I = \int_{0}^{2026} \frac{x^5}{x^5 + (2026 - x)^5} \, dx\).
Direct algebraic integration is not practical, which suggests using definite integral symmetry properties.
Step 2: Key Formula or Approach:
We use King's Property of definite integrals:
\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx \]
Applying this with \(a = 0\) and \(b = 2026\), we replace \(x\) with \(2026 - x\).
Step 3: Detailed Explanation:
1. Let the given integral be \(I\):
\[ I = \int_{0}^{2026} \frac{x^5}{x^5 + (2026 - x)^5} \, dx \quad \cdots (1) \]
2. Apply King's Property by replacing \(x\) with \(2026 - x\) in the integrand:
\[ I = \int_{0}^{2026} \frac{(2026 - x)^5}{(2026 - x)^5 + (2026 - (2026 - x))^5} \, dx \]
3. Simplify the denominator terms:
\[ I = \int_{0}^{2026} \frac{(2026 - x)^5}{(2026 - x)^5 + x^5} \, dx \quad \cdots (2) \]
4. Since Equation (1) and Equation (2) have identical limits, we add them together:
\[ 2I = \int_{0}^{2026} \left[ \frac{x^5}{x^5 + (2026 - x)^5} + \frac{(2026 - x)^5}{x^5 + (2026 - x)^5} \right] dx \]
\[ 2I = \int_{0}^{2026} \frac{x^5 + (2026 - x)^5}{x^5 + (2026 - x)^5} \, dx \]
5. The integrand simplifies to 1:
\[ 2I = \int_{0}^{2026} 1 \cdot dx \]
6. Evaluate the integral:
\[ 2I = \Big[ x \Big]_{0}^{2026} = 2026 - 0 = 2026 \]
\[ I = \frac{2026}{2} = 1013 \]
This matches Option (B).
Step 4: Final Answer:
The value of the definite integral is \(1013\), which corresponds to Option (B).