Integral of \( \displaystyle \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} \)
Problem 4.922 · medium
Find \( \displaystyle \int \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(2 x - 1\right) \ln{\left(2 x - 1 \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \frac{\left(2 x - 1\right)^{2} \ln{\left(2 x - 1 \right)}}{4} - \int \left(x - \frac{1}{2}\right)\, dx \]parts simplifyUse integration by parts with u = log(2*x - 1) and dv = (2*x - 1)dx. Simplify the integrand in the remaining integral.✓ Proved
- \[ = - \frac{x^{2}}{2} + \frac{x}{2} + \frac{\left(2 x - 1\right)^{2} \ln{\left(2 x - 1 \right)}}{4} \]antiderivative simplifyEvaluate the integral of the remaining term. Distribute and simplify the expression.✓ Proved
Answer \( - \frac{x^{2}}{2} + \frac{x}{2} + \left(x^{2} - x\right) \ln{\left(2 x - 1 \right)} + \frac{\ln{\left(2 x - 1 \right)}}{4} + C \)
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. 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ 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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final expression omits the +\u00bc\,\log(2x-1) term that arises from distributing (2x-1)^2/4 over the logarithm. The correct antiderivative is \u00b1xqwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: inconclusive 2026-10-12 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final expression omits the +\u00bc\,\log(2x-1) term that arises from distributing (2x-1)^2/4 over the logarithm. The correct antiderivative is \u00b1xqwen3.6:27b-mlx: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies integration by parts and simplifies the result. The final expression is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-10-12
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 2026-10-12 with SymPy 1.14.0.