Question

\(\int_{0}^{2} \frac{x^{4}}{x^{4} + (2 - x)^{4}} \, dx =\)

• A. 1
• B. 2
• C. 4
• D. 8
• E. 0

Hint:

Property: \(\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx\). Add the two expressions.

Answer 1

The Correct Option is A

Step 1: Concept:
• Use the property: \[ \int_{0}^{a} f(x)\,dx = \int_{0}^{a} f(a-x)\,dx \]
• Here, \(a = 2\).

Step 2: Detailed Explanation:
• Let: \[ I = \int_{0}^{2} \frac{x^4}{x^4 + (2-x)^4}\,dx \]
• Using the property: \[ I = \int_{0}^{2} \frac{(2-x)^4}{(2-x)^4 + x^4}\,dx \]
• Add both expressions: \[ 2I = \int_{0}^{2} \frac{x^4 + (2-x)^4}{x^4 + (2-x)^4}\,dx \]
• Simplify: \[ 2I = \int_{0}^{2} 1\,dx = 2 \]
• Hence: \[ I = 1 \]

Step 3: Final Answer:
• The value is \(1\).