Question

Evaluate the integral: \[ \int_{\frac{\pi}{10}}^{\frac{2\pi}{5}} \frac{\cot^3 x}{1 + \cot^3 x} \, dx \]

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{3} \)
• C. \( 0 \)
• D. \( \frac{1}{4} \)

Hint:

For integrals involving trigonometric functions, always check for symmetry. Symmetric intervals often lead to the integral being zero, especially for odd functions.

Answer 1

The Correct Option is C

We are tasked with evaluating the integral: \[ I = \int_{\frac{\pi}{10}}^{\frac{2\pi}{5}} \frac{\cot^3 x}{1 + \cot^3 x} \, dx \]

1. Step 1: Use symmetry in trigonometric integrals We can observe that \( \cot^3 x \) has a symmetry when integrated over a certain range. The integrand \( \frac{\cot^3 x}{1 + \cot^3 x} \) is symmetric over the interval from \( \frac{\pi}{10} \) to \( \frac{2\pi}{5} \), meaning that the integral results in zero because the function is odd with respect to its midpoint.

2. Step 2: Understanding the symmetry Since the function behaves symmetrically, the integral evaluates to zero over this symmetric interval. Thus, the answer is: \[ I = 0 \]