Evaluate: ∫_(0 to 1) [tan^(-1) x] dx.
Answer and explanation
Correct answer: C. (π / 4) - (1/2) ln 2
- Step 1: Integration by parts: ∫ 1 * tan^(-1) x dx = x tan^(-1) x - ∫ [x / (1 + x^2)] dx.
- Step 2: = x tan^(-1) x - (1/2) ln(1 + x^2).
- Step 3: Evaluated from 0 to 1: [1 * tan^(-1)(1) - (1/2) ln(2)] - [0 - 0] = π/4 - (1/2) ln 2.
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