Evaluate: ∫_(0 to π) [x / (1 + sin x)] dx.
Answer and explanation
Correct answer: A. π
- Step 1: Using King's property: I = ∫_(0 to π) [(π - x) / (1 + sin(π - x))] dx = ∫_(0 to π) [(π - x) / (1 + sin x)] dx.
- Step 2: 2I = π ∫_(0 to π) [1 / (1 + sin x)] dx.
- Step 3: Multiply numerator and denominator by 1 - sin x: ∫_(0 to π) (1 - sin x) / cos^2 x dx = [tan x - sec x]_(0 to π).
- Step 4: (tan π - sec π) - (tan 0 - sec 0) = (0 - (-1)) - (0 - 1) = 1 + 1 = 2.
- Step 5: 2I = π * 2 => I = π.
Why not the others: The alternative choices (π / 2, 2π, π / 4) stem from algebraic sign mistakes, misapplying coordinate formulas, or intermediate arithmetic calculation slips.
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