Question:
Evaluate \( \int_0^1 \log \left( \frac{1}{x - 1} \right) \, dx \):
Evaluate \( \int_0^1 \log \left( \frac{1}{x - 1} \right) \, dx \):
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For integrals involving logarithms, simplify the integrand using logarithmic properties and analyze the behavior of the function at the limits of integration.
Updated On: Apr 7, 2026
- \( \frac{1}{4} \)
- \( \frac{1}{2} \)
- \( -1 \)
- \( 3 \)
- \( 0 \)
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The Correct Option is
Solution and Explanation
The given integral can be evaluated using logarithmic properties and integration techniques.
We rewrite the logarithmic term: \[ \log \left( \frac{1}{x - 1} \right) = - \log(x - 1) \] Thus, the integral becomes: \[ \int_0^1 \log \left( \frac{1}{x - 1} \right) dx = - \int_0^1 \log(x - 1) \, dx \] The integral of \( \log(x - 1) \) from 0 to 1 gives 0, as the value of the integral at these limits cancels out due to symmetry.
Thus, the final answer is 0.
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