What is \int_{0}^{1}\ln(\frac{1}{x}-1)\,dx equal to?
- A. -1
- B. 0 ✓
- C. 1
- D. \ln 2
Correct Answer: B. 0
Explanation
Let I = \int_{0}^{1}\ln(\frac{1-x}{x})\,dx. Using the property \int_{0}^{a}f(x)\,dx = \int_{0}^{a}f(a-x)\,dx, substitute x with 1-x. Then I = \int_{0}^{1}\ln(\frac{x}{1-x})\,dx = \int_{0}^{1}-\ln(\frac{1-x}{x})\,dx = -I. This implies 2I = 0, so I = 0.
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