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Evaluate the definite integral ∫₀^(π) √(1+4sin² ((x)/(2))-4sin ((x)/(2))) dx:

1. 4√(3)-4
2. 4√(3)-4-(π)/(3)
3. π-4
4. (2π)/(3)-4-4√(3)

Correct answer: 4√(3)-4-(π)/(3)

Solution

The correct option is derived from evaluating the integral by simplifying the integrand using trigonometric identities and then applying integration techniques, which ultimately leads to the result that includes both a constant term and a term involving pi, confirming the complexity of the integral.

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