Evaluate the definite integrals`int_0^pi (xtanx)/(secx+tanx)dx`

Explanation

To evaluate the definite integral \( I = \int_0^\pi \frac{x \tan x}{\sec x + \tan x} \, dx \), we will use the property of definite integrals and some algebraic manipulations. Here’s the step-by-step solution:

### Step 1: Use the property of definite integrals
We know that:
\[ \int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx \]
For our integral, we can set \( a = \pi \):
...