The integral `int_0^pisqrt(1+4sin^2(x/2)-4sin(x/2)dx)` equal (1) `pi-4` (2) `(2pi)/3-4-4sqrt(3)` (3) `4""sqrt(3)-4` (4) `4""sqrt(3)-4-pi/3`

Explanation

To solve the integral \( I = \int_0^{\pi} \sqrt{1 + 4 \sin^2\left(\frac{x}{2}\right) - 4 \sin\left(\frac{x}{2}\right)} \, dx \), we will follow these steps:

### Step 1: Simplify the expression under the square root
We start by rewriting the expression inside the square root:
\[ 1 + 4 \sin^2\left(\frac{x}{2}\right) - 4 \sin\left(\frac{x}{2}\right) = (1 - 2 \sin\left(\frac{x}{2}\right))^2 \]
This is because:
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