Question

$\int_{0}^{\pi/2} \frac{\sin^{100}x}{\sin^{100}x + \cos^{100}x} dx =$

• A. $\pi/2$
• B. $\pi/4$
• C. 100
• D. 50

Hint:

For symmetric integrals of the form $\frac{f(\sin)}{f(\sin)+f(\cos)}$ from $0$ to $\pi/2$, the answer is always $(Upper Limit - Lower Limit) / 2$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a definite integral that uses the property \(\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx\). This property allows us to "swap" \(\sin\) and \(\cos\) terms in the interval \([0, \pi/2]\).
Step 2: Key Formula or Approach:
1. Let the integral be \(I\).
2. Apply property: Replace \(x\) with \(\pi/2 - x\).
3. Use \(\sin(\pi/2 - x) = \cos x\) and \(\cos(\pi/2 - x) = \sin x\).
Step 3: Detailed Explanation:
Let \(I = \int_{0}^{\pi/2} \frac{\sin^{100} x}{\sin^{100} x + \cos^{100} x} \, dx \). Using the property: \[ I = \int_{0}^{\pi/2} \frac{\sin^{100} (\pi/2 - x)}{\sin^{100} (\pi/2 - x) + \cos^{100} (\pi/2 - x)} \, dx = \int_{0}^{\pi/2} \frac{\cos^{100} x}{\cos^{100} x + \sin^{100} x} \, dx \] Add the two expressions for \(I\): \[ 2I = \int_{0}^{\pi/2} \frac{\sin^{100} x + \cos^{100} x}{\sin^{100} x + \cos^{100} x} \, dx \] \[ 2I = \int_{0}^{\pi/2} 1 \, dx = [x]_0^{\pi/2} = \pi/2 \] \[ I = \frac{\pi/2}{2} = \pi/4 \]
Step 4: Final Answer:
The value of the definite integral is \( \pi/4 \).