Step 1: Understanding the Question:
This is a standard definite integral problem involving trigonometric functions.
The structure of the integrand suggests the use of periodic or symmetric properties of definite integrals.
Step 2: Key Formula or Approach:
We use the property:
\[ \int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx \]
This is often called the "King's Property" in calculus.
Step 3: Detailed Explanation:
Let \( I = \int_{0}^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} \, dx \quad \text{--- (1)} \)
Applying the property \( x \to \frac{\pi}{2} - x \):
\[ I = \int_{0}^{\pi/2} \frac{\sin^n(\frac{\pi}{2} - x)}{\sin^n(\frac{\pi}{2} - x) + \cos^n(\frac{\pi}{2} - x)} \, dx \]
Since \( \sin(\frac{\pi}{2} - x) = \cos x \) and \( \cos(\frac{\pi}{2} - x) = \sin x \):
\[ I = \int_{0}^{\pi/2} \frac{\cos^n x}{\cos^n x + \sin^n x} \, dx \quad \text{--- (2)} \]
Adding equations (1) and (2):
\[ 2I = \int_{0}^{\pi/2} \frac{\sin^n x + \cos^n x}{\sin^n x + \cos^n x} \, dx \]
\[ 2I = \int_{0}^{\pi/2} 1 \, dx \]
\[ 2I = [x]_0^{\pi/2} = \frac{\pi}{2} - 0 \]
\[ I = \frac{\pi}{4} \]
Step 4: Final Answer:
The value of the integral is \( \frac{\pi}{4} \).