Integral of \( \displaystyle \left(3 x - 1\right)^{2} \ln{\left(3 x - 1 \right)} \)
Problem 4.1248 · medium Video
Find \( \displaystyle \int \left(3 x - 1\right)^{2} \ln{\left(3 x - 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(3 x - 1\right)^{2} \ln{\left(3 x - 1 \right)}\, dx \]integral rewriteStart with the integral of the given function. The integrand is already in a form suitable for integration by parts.✓ Proved
- \[ = \frac{\left(3 x - 1\right)^{3} \ln{\left(3 x - 1 \right)}}{9} - \int \frac{\left(3 x - 1\right)^{2}}{3}\, dx \]parts algebraApply integration by parts with u = log(3*x - 1) and dv = (3*x - 1)**2 dx. Simplify the integrand in the remaining integral.✓ Proved
- \[ = \frac{\left(3 x - 1\right)^{3} \ln{\left(3 x - 1 \right)}}{9} - \frac{\left(3 x - 1\right)^{3}}{27} \]antiderivativeIntegrate the remaining polynomial term.✓ Proved
- \[ = \frac{\left(3 x - 1\right)^{3} \left(3 \ln{\left(3 x - 1 \right)} - 1\right)}{27} \]simplifyCombine the terms into a single expression.✓ Proved
Answer \( - x^{3} + x^{2} - \frac{x}{3} + \left(3 x^{3} - 3 x^{2} + x\right) \ln{\left(3 x - 1 \right)} - \frac{\ln{\left(3 x - 1 \right)}}{9} + C \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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 |
| 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 | lines differ by the constant -1/27 log is undefined for non-positive arguments |
| 6 | ✓ 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: lines differ by the constant 1/27 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: passqwen3.6:27b-mlx: pass — The solution correctly applies integration by parts and simplifies the result. Each step applies a single rule and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies integration by parts and simplifies the result. Each step applies a single rule and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies integration by parts and simplifies the result. The final answer is algebraically equivalent to the stated answer, and each step adheres to the single-rule constraint.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies integration by parts and simplifies the result. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (style) 2026-10-16 — Step 3 applies integration by parts and immediately simplifies the integrand in a single step, violating the rule that each step must change only one thing. It should be split into a parts step followed by an algebraic simplification step.
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 2026-10-19 with SymPy 1.14.0.