Question

Evaluate \[ \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\tan x}{\tan x + \cot x} \, dx \]

• A. \( \frac{\pi}{2} \)
• B. \( \frac{3\pi}{10} \)
• C. \( \frac{\pi}{5} \)
• D. \( \frac{\pi}{20} \)

Hint:

When encountering integrals with trigonometric functions, try to simplify the integrand using trigonometric identities before proceeding with the integration.

Answer 1

The Correct Option is D

Step 1: Simplify the integrand.
The given integral has the form \( \frac{\tan x}{\tan x + \cot x} \). We can simplify the expression in the integrand by applying the identity \( \cot x = \frac{1}{\tan x} \), which results in a much simpler integral.

Step 2: Perform the integration.
After simplifying, we integrate the function from \( \frac{\pi}{5} \) to \( \frac{3\pi}{10} \). The result of the integration is \( \frac{\pi}{20} \).

Step 3: Conclusion.
Thus, the correct answer is \( \frac{\pi}{20} \), corresponding to option (D).