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

Step 1: Understanding the Concept:
In calculus, "King's Property" is a fundamental tool for solving definite integrals, especially those involving trigonometric ratios.
This property utilizes the symmetry of functions over an interval.
By replacing the variable with the sum of the limits minus the variable, the form of the integrand changes while the integral's value remains constant.
This transformation often allows us to create a second equation that, when added to the first, simplifies the expression significantly, often resulting in a constant integrand.
Step 2: Key Formula or Approach:
The integral property is defined as:
\[ I = \int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx \]
For limits \(0\) to \(\frac{\pi}{2}\), substituting \(x \to \frac{\pi}{2} - x\) transforms sine into cosine and cosine into sine.
Step 3: Detailed Explanation:
Let the initial integral be denoted by \(I\):
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^5 x}{\sin^5 x + \cos^5 x} dx \quad \dots (1) \]
Now, apply King's Property by replacing \(x\) with \((\frac{\pi}{2} - x)\):
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^5 (\frac{\pi}{2} - x)}{\sin^5 (\frac{\pi}{2} - x) + \cos^5 (\frac{\pi}{2} - x)} dx \]
Using the trigonometric co-function identities:
\(\sin(\frac{\pi}{2} - x) = \cos x\)
\(\cos(\frac{\pi}{2} - x) = \sin x\)
The integral becomes:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\cos^5 x}{\cos^5 x + \sin^5 x} dx \quad \dots (2) \]
Since the limits of integration are the same for both equations (1) and (2), we can add them together:
\[ I + I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^5 x}{\sin^5 x + \cos^5 x} dx + \int_{0}^{\frac{\pi}{2}} \frac{\cos^5 x}{\sin^5 x + \cos^5 x} dx \]
\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{\sin^5 x + \cos^5 x}{\sin^5 x + \cos^5 x} \right) dx \]
The expression inside the integral simplifies to \(1\):
\[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \cdot dx \]
Integrating with respect to \(x\):
\[ 2I = [x]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2} \]
Divide by 2 to solve for \(I\):
\[ I = \frac{\pi}{4} \]
Step 4: Final Answer:
The result of the definite integral is \(\frac{\pi}{4}\), confirming option (A).