Question

Evaluate the integral \[ \int_{0}^{\pi/2} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x}\, dx \]

• A. \( \dfrac{\pi}{4} \)
• B. \( \dfrac{\pi}{8} \)
• C. \( \dfrac{\pi}{2} \)
• D. \( \pi \)

Hint:

When an integral has symmetric limits and complementary trigonometric terms, try using the substitution \( x \to \frac{\pi}{2}-x \).

Answer 1

The Correct Option is A

Step 1: Use the property of definite integrals.
Let \[ I = \int_{0}^{\pi/2} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x}\, dx \] Using the property \( I = \int_{0}^{\pi/2} f(x)\,dx = \int_{0}^{\pi/2} f\!\left(\frac{\pi}{2}-x\right)\,dx \).

Step 2: Replace \( x \) by \( \frac{\pi}{2} - x \).
\[ I = \int_{0}^{\pi/2} \frac{\cos^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x}\, dx \]

Step 3: Add both expressions.
\[ 2I = \int_{0}^{\pi/2} 1 \, dx \] \[ 2I = \frac{\pi}{2} \]

Step 4: Conclusion.
\[ I = \boxed{\frac{\pi}{4}} \]