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Evaluate the definite integral I = integral from 0 to 1/2 of [ln(1 + 2x) / (1 + 4x²)] dx.

1. pi*ln(2)/8
2. pi*ln(2)/4
3. pi*ln(2)/32
4. pi*ln(2)/16

Correct answer: pi*ln(2)/16

Solution

Substituting 2x = tan(theta) transforms I into (1/2)*integral from 0 to pi/4 of ln(1+tan(theta)) d(theta). Using the standard result (pi/8)*ln2 gives I = pi*ln(2)/16.

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