Question

Evaluate \( \int_0^1 \log \left( \frac{1}{x - 1} \right) \, dx \):

• A. \( \frac{1}{4} \)
• B. \( \frac{1}{2} \)
• C. \( -1 \)
• D. \( 3 \)
• E. \( 0 \)

Hint:

For integrals involving logarithms, simplify the integrand using logarithmic properties and analyze the behavior of the function at the limits of integration.

Answer 1

Answer: (E)

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.