Integral of \( \displaystyle \left(3 x + 1\right)^{2} \ln{\left(3 x + 1 \right)} \)
Problem 4.1475 · medium Video
Find \( \displaystyle \int \left(3 x + 1\right)^{2} \ln{\left(3 x + 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(3 x + 1\right)^{2} \ln{\left(3 x + 1 \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(9 x^{2} + 6 x + 1\right) \ln{\left(3 x + 1 \right)}\, dx \]algebraExpand the squared term.✓ Proved
- \[ = 6 \int x \ln{\left(3 x + 1 \right)}\, dx + 9 \int x^{2} \ln{\left(3 x + 1 \right)}\, dx + \int \ln{\left(3 x + 1 \right)}\, dx \]linearityDistribute the integral over the sum.✓ Proved
- \[ = \int \left(3 x + 1\right)^{2} \ln{\left(3 x + 1 \right)}\, dx \]partsUse integration by parts on the original form.✓ 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 simplifyApply integration by parts with u = log(3*x + 1) and dv = (3*x + 1)**2 dx. Simplify the integrand in the remaining integral.✓ Proved
- \[ = \frac{\left(3 x + 1\right)^{3} \ln{\left(3 x + 1 \right)}}{9} - \frac{\int \left(3 x + 1\right)^{2}\, dx}{3} \]linearityFactor out the constant.✓ Proved
- \[ = \frac{\left(3 x + 1\right)^{3} \ln{\left(3 x + 1 \right)}}{9} - \frac{\left(3 x + 1\right)^{3}}{27} \]antiderivative simplifyEvaluate the integral. Simplify the final expression.✓ Proved
Answer \( - x^{3} - x^{2} - \frac{x}{3} + \left(3 x^{3} + 3 x^{2} + x\right) \ln{\left(3 x + 1 \right)} + \frac{\ln{\left(3 x + 1 \right)}}{9} + C \)
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. 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 | simplify(a - b) reduced to 0 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 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 8 | ✓ Proved | sympy 1.14.0 | lines differ by the constant 1/27 log is undefined for non-positive arguments |
| 9 | ✓ 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: fail (style) — The solution introduces an unnecessary and incorrect step 4 that repeats the original integral and claims to use integration by parts, yet it does not actually apply the rule or progress the calculation. This violates the rule that each step must change only one thing and be properly labeled.qwen3.6:27b-mlx: pass — The solution correctly applies integration by parts to the original integrand. Although steps 2 and 3 are unused dead ends, they are algebraically correct and do not constitute a defect in the logic of the final result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-09 — The solution correctly applies integration by parts to the original integrand. Although steps 2 and 3 are unused dead ends, they are algebraically correct and do not constitute a defect in the logic of the final result.gpt-oss:20b: fail (style) 2026-11-09 — The solution introduces an unnecessary and incorrect step 4 that repeats the original integral and claims to use integration by parts, yet it does not actually apply the rule or progress the calculation. This violates the rule that each step must change only one thing and be properly labeled.qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly applies integration by parts to the original integrand, simplifies the resulting expression, and arrives at the correct antiderivative. Each step applies a single rule and uses valid labels from the vocabulary.gpt-oss:20b: fail (error) 2026-11-08 — Step 4 merely restates the original integral without applying any rule; it introduces an unnecessary and incorrect step. The solution should proceed directly to the integration‑by‑parts step in step 5.qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly applies integration by parts to the original integrand. Although steps 2 and 3 are abandoned, they do not constitute a defect as the subsequent steps (4-9) form a valid, self-contained derivation of the correct result. Each step applies a single rule from the allowed vocabulary.gpt-oss:20b: fail (error) 2026-11-08 — Step 4 claims to apply integration by parts but the expression remains exactly the same as the original integral, so no rule was actually applied. This violates the rule that each step must change only one thing and must correctly label the applied rule.
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 2026-11-09 with SymPy 1.14.0.