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Evaluate the definite integral: I = integral from pi/3 to pi/2 of (2 + 3*sin(x)) / (sin(x) * (1 + cos(x))) dx

1. 7/2 - sqrt(3) - logₑ(sqrt(3))
2. -2 + 3*sqrt(3) + logₑ(sqrt(3))
3. 10/3 - sqrt(3) + logₑ(sqrt(3))
4. 10/3 - sqrt(3) - logₑ(sqrt(3))

Correct answer: 7/2 - sqrt(3) - logₑ(sqrt(3))

Solution

After splitting and substituting t=tan(x/2), the integral reduces to elementary pieces. Evaluating from x=pi/3 (t=1/sqrt(3)) to x=pi/2 (t=1) yields 7/2 - sqrt(3) - logₑ(sqrt(3)).

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