Integral of \( \displaystyle \frac{\ln{\left(2 x - 1 \right)}}{2 x - 1} \)
Problem 4.1493 · medium Video
Find \( \displaystyle \int \frac{\ln{\left(2 x - 1 \right)}}{2 x - 1} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{\ln{\left(2 x - 1 \right)}}{2 x - 1}\, dx \]integral rewriteStart with the integral of the given function. Rewrite the fraction using a negative exponent.✓ Proved
- \[ = - \int 0\, dx + \int \frac{\ln{\left(2 x - 1 \right)}}{2 x - 1}\, dx \]substitutionWe will prepare for substitution by noting the derivative of the inner function.✓ Proved
- \[ = \frac{\int \frac{2 \ln{\left(2 x - 1 \right)}}{2 x - 1}\, dx}{2} \]algebra algebraMultiply and divide by 2 to form the derivative of (2x-1) inside the integral. This is a step to prepare for the substitution logic without introducing new variables.✓ Proved
- \[ = \frac{\ln{\left(2 x - 1 \right)}^{2}}{4} \]antiderivative simplifyThe antiderivative of log(v)/v is log(v)^2 / 2. Simplify the expression to the final result.✓ Proved
Answer \( \frac{\ln{\left(2 x - 1 \right)}^{2}}{4} + C \)
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. A reviewer's objection to the explanation stands; see the receipt. How we verify
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 2*x - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 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 (error) — Step 5 multiplies the integrand by an extra factor of 2 (and a trivial (2*x-1)**0), changing the integrand and applying two operations at once. Additionally, step 3’s “substitution” label is misleading because no substitution is actually performed.qwen3.6:27b-mlx: fail (error) — Step 3 is mathematically invalid; subtracting the integral of 0 is not a valid application of the substitution rule, and the step does not perform a substitution. Step 4 incorrectly labels algebraic manipulation (multiplying by 1) as 'algebra' while implicitly relying on substitution logic that was not properly established, violating the one-rule-per-step constraint by mixing preparation for substitution with algebraic identity.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-13 — Step 3 is mathematically invalid; subtracting the integral of 0 is not a valid application of the substitution rule, and the step does not perform a substitution. Step 4 incorrectly labels algebraic manipulation (multiplying by 1) as 'algebra' while implicitly relying on substitution logic that was not properly established, violating the one-rule-per-step constraint by mixing preparation for substitution with algebraic identity.gpt-oss:20b: fail (error) 2026-11-13 — Step 5 multiplies the integrand by an extra factor of 2 (and a trivial (2*x-1)**0), changing the integrand and applying two operations at once. Additionally, step 3’s “substitution” label is misleading because no substitution is actually performed.qwen3.6:27b-mlx: fail (error) 2026-11-12 — Step 3 is mathematically invalid; subtracting the integral of 0 does not constitute a substitution, and the label 'substitution' is incorrect for this operation. Step 4 also mislabels the introduction of a constant factor as 'algebra' when it is part of the substitution process, but the primary defect is the nonsensical Step 3.gpt-oss:20b: fail (error) 2026-11-12 — Step 6 applies the antiderivative to the integrand 2·log(v)/v, whose integral is log(v)^2, but the solution writes 1/2·(log(v)^2/2), effectively halving the correct result. This mis‑applies the antiderivative rule and yields an incorrect factor of 1/4 instead of 1/2.qwen3.6:27b-mlx: fail (error) 2026-11-12 — Step 3 is mathematically nonsensical; subtracting the integral of 0 does not constitute a valid substitution step, and the label 'substitution' is incorrect for this operation. Furthermore, Step 6 applies the antiderivative rule incorrectly by failing to account for the chain rule factor of 2 from the inner function (2x-1), effectively skipping the necessary substitution logic that was not properly established in previous steps.gpt-oss:20b: fail (error) 2026-11-12 — Step 6 applies the antiderivative of log(v)/v incorrectly: the integrand actually contains an extra factor of 2, so the result should be (log(2*x-1))^2/2 before the outer 1/2, yielding the final answer log(2*x-1)^2/4. The step combines two rules (substitution and antiderivative) in one line and mislabels the operation.
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 2026-11-13 with SymPy 1.14.0.