Question

The value of $ \int_0^{\frac{\pi}{2}} \frac{\tan x}{\tan x + \cot x} \, dx \text{ is} $

• A. 0
• B. \( \frac{\pi}{2} \)
• C. \( \frac{\pi}{4} \)
• D. None of these

Hint:

Use trigonometric identities to simplify the integrand, and recognize standard integral results when possible.

Answer 1

The Correct Option is C

We are tasked with evaluating the integral: \[ I = \int_0^{\frac{\pi}{2}} \frac{\tan x}{\tan x + \cot x} \, dx \]
Step 1: Simplify the integrand
We can use the identity \( \cot x = \frac{1}{\tan x} \), so the integrand becomes: \[ \frac{\tan x}{\tan x + \frac{1}{\tan x}} = \frac{\tan^2 x}{\tan^2 x + 1} \]
Step 2: Use trigonometric identity
Now use the identity \( 1 + \tan^2 x = \sec^2 x \), so the integrand simplifies to: \[ \frac{\tan^2 x}{\sec^2 x} = \sin^2 x \]
Step 3: Perform the integral
The integral of \( \sin^2 x \) over the interval from 0 to \( \frac{\pi}{2} \) is known to be \( \frac{\pi}{4} \).
Step 4: Conclusion
Thus, the value of the integral is \( \frac{\pi}{4} \), corresponding to option (c).