Integral of \( \displaystyle \left(x + 1\right)^{2} \ln{\left(x + 1 \right)} \)
Problem 4.1321 · medium Video
Find \( \displaystyle \int \left(x + 1\right)^{2} \ln{\left(x + 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(x + 1\right)^{2} \ln{\left(x + 1 \right)}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = \int \left(x^{2} + 2 x + 1\right) \ln{\left(x + 1 \right)}\, dx \]algebraExpand the squared term.✓ Proved
- \[ = \int 2 x \ln{\left(x + 1 \right)}\, dx + \int x^{2} \ln{\left(x + 1 \right)}\, dx + \int \ln{\left(x + 1 \right)}\, dx \]linearityDistribute the logarithm over the sum.✓ Proved
- \[ = 2 \int x \ln{\left(x + 1 \right)}\, dx + \int x^{2} \ln{\left(x + 1 \right)}\, dx + \int \ln{\left(x + 1 \right)}\, dx \]linearityPull out the constant factor 2.✓ Proved
- \[ = \int \left(x + 1\right)^{2} \ln{\left(x + 1 \right)}\, dx \]simplifyRecognize the original integrand structure.✓ Proved
- \[ = \frac{\left(x + 1\right)^{3} \ln{\left(x + 1 \right)}}{3} - \int \frac{\left(x + 1\right)^{2}}{3}\, dx \]parts algebraApply integration by parts with u = (x+1)**2 * log(x+1) is not efficient, let's use u = log(x+1) and dv = (x+1)**2 dx. Simplify the integrand in the remaining integral.✓ Proved
- \[ = \frac{\left(x + 1\right)^{3} \ln{\left(x + 1 \right)}}{3} - \frac{\left(x + 1\right)^{3}}{9} \]antiderivativeEvaluate the integral of the power function.✓ Proved
- \[ = - \frac{x^{3}}{9} - \frac{x^{2}}{3} - \frac{x}{3} + \frac{\left(x + 1\right)^{3} \ln{\left(x + 1 \right)}}{3} - \frac{1}{9} \]algebra algebraExpand the cubic term. Distribute the denominator 9.✓ Proved
Answer \( - \frac{x^{3}}{9} - \frac{x^{2}}{3} - \frac{x}{3} + \left(\frac{x^{3}}{3} + x^{2} + x\right) \ln{\left(x + 1 \right)} + \frac{\ln{\left(x + 1 \right)}}{3} + C \)
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 1/9 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 |
| 10 | ✓ 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 -1/9 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 3 incorrectly claims to distribute the logarithm over the sum; the logarithm is not a linear operator. Step 5 reverts to the original integrand without justification, violating the one‑rule‑per‑step rule. The final antiderivative also omits the required +log(x+1)/3 term and includes an extraneous constant –1/9, indicating a misapplication of integration by parts and algebraic simplification.qwen3.6:27b-mlx: fail (error) — Step 5 is a logical error: it reverts the expression to the original integral, creating a circular reference that breaks the derivation chain. Additionally, Step 6 applies integration by parts to the entire expression from Step 5, but the label 'parts' is applied to a step that also implicitly relies on the structure established in previous steps, making the transition from Step 4 to Step 5 to Step 6 disjointed and logically invalid as a linear proof.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-23 — Step 5 is a logical error: it reverts the expression to the original integral, creating a circular reference that breaks the derivation chain. Additionally, Step 6 applies integration by parts to the entire expression from Step 5, but the label 'parts' is applied to a step that also implicitly relies on the structure established in previous steps, making the transition from Step 4 to Step 5 to Step 6 disjointed and logically invalid as a linear proof.gpt-oss:20b: fail (error) 2026-10-23 — Step 3 incorrectly claims to distribute the logarithm over the sum; the logarithm is not a linear operator. Step 5 reverts to the original integrand without justification, violating the one‑rule‑per‑step rule. The final antiderivative also omits the required +log(x+1)/3 term and includes an extraneous constant –1/9, indicating a misapplication of integration by parts and algebraic simplification.qwen3.6:27b-mlx: fail (error) 2026-10-22 — Step 5 reverts the expression to the original integral without performing any operation, breaking the 'one change per step' rule and creating a logical loop. Step 6 applies integration by parts to the entire integral from Step 5, but the transition from Step 4 (a sum of three integrals) to Step 5 (the combined integral) is unjustified and redundant, making the flow incoherent.gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: fail (error) 2026-10-22 — Step 5 is a logical error: it reverts the expression to the original integral without performing any integration, breaking the chain of derivation. Furthermore, Step 6 applies integration by parts to the original integral (from Step 5) rather than continuing from the split terms in Step 4, making the previous steps irrelevant and the transition invalid.gpt-oss:20b: fail (style) 2026-10-22 — Step 5 re‑introduces the original integrand without a valid justification, and step 3’s note incorrectly describes the operation as “distributing the logarithm over the sum.” Both steps violate the rule‑granularity requirement.
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 2026-10-23 with SymPy 1.14.0.