Integrate the functions`tan^(-1)sqrt((1-x)/(1+x))`

Explanation

To solve the integral \( I = \int \tan^{-1} \left( \sqrt{\frac{1-x}{1+x}} \right) dx \), we will follow a series of steps involving substitution and integration by parts.

### Step 1: Substitution
Let \( x = \cos(2\theta) \). Then, we differentiate both sides:
\[ dx = -2 \sin(2\theta) d\theta \]

...