Integral of \( \displaystyle \frac{\ln{\left(x - 1 \right)}}{x - 1} \)
Problem 4.562 · medium
Find \( \displaystyle \int \frac{\ln{\left(x - 1 \right)}}{x - 1} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{\ln{\left(x - 1 \right)}}{x - 1}\, dx \]integral algebraStart with the integral of the given function. Rewrite the fraction using a negative exponent.✓ Proved
- \[ = \frac{\ln{\left(x - 1 \right)}^{2}}{2} \]antiderivativeApply the power rule for the logarithm via substitution u = log(x - 1).✓ Proved
Answer \( \frac{\ln{\left(x - 1 \right)}^{2}}{2} + C \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: fail (style) — Step 3 applies two rules at once (substitution and power rule) in a single step, violating the one‑change‑per‑step rule.qwen3.6:27b-mlx: fail (style) — Step 3 applies substitution and integration in a single step, violating the 'one rule per step' constraint. It should be split into a substitution step and an integration step.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-07 — Step 3 applies two rules at once (substitution and power rule) in a single step, violating the one‑change‑per‑step rule.qwen3.6:27b-mlx: fail (style) 2026-10-07 — Step 3 applies substitution and integration in a single step, violating the 'one rule per step' constraint. It should be split into a substitution step and an integration step.qwen3.6:27b-mlx: fail (style) 2026-10-07 — Step 3 applies substitution (u = log(x-1)) to transform the integral, but labels it 'antiderivative'. The label 'substitution' exists in the vocabulary and is the correct rule for this step. Additionally, the note claims to apply the 'power rule for the logarithm', which is mathematically imprecise; it applies the power rule for integration to the substituted variable.gpt-oss:20b: fail (error) 2026-10-07 — Step 3 applies two rules at once (substitution and antiderivative) and is labeled incorrectly as "antiderivative". Each step must change only one thing and use the correct rule name.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-10-07 with SymPy 1.14.0.