Integral of \( \displaystyle \left(x + 1\right) \ln{\left(x + 1 \right)} \)
Problem 4.651 · medium
Find \( \displaystyle \int \left(x + 1\right) \ln{\left(x + 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(x + 1\right) \ln{\left(x + 1 \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \left(\frac{x^{2}}{2} + x\right) \ln{\left(x + 1 \right)} - \int \frac{\frac{x^{2}}{2} + x}{x + 1}\, dx \]partsUse integration by parts with u = log(x + 1) and dv = (x + 1)dx.✓ Proved
- \[ = \left(\frac{x^{2}}{2} + x\right) \ln{\left(x + 1 \right)} - \int \frac{x \left(x + 2\right)}{2 x + 2}\, dx \]algebra algebraSimplify the integrand by factoring x from the numerator. Rewrite the term (x + 2) to facilitate division.✓ Proved
- \[ = \left(\frac{x^{2}}{2} + x\right) \ln{\left(x + 1 \right)} - \int \left(\frac{x}{2} + \frac{x}{2 \left(x + 1\right)}\right)\, dx \]algebraDistribute the numerator terms.✓ Proved
- \[ = \left(\frac{x^{2}}{2} + x\right) \ln{\left(x + 1 \right)} - \int \left(\frac{x}{2} + \frac{x}{2 x + 2}\right)\, dx \]algebra algebraSplit the fraction. Rewrite the numerator of the second term.✓ Proved
- \[ = \left(\frac{x^{2}}{2} + x\right) \ln{\left(x + 1 \right)} - \int \left(\frac{x}{2} + \frac{1}{2} - \frac{1}{2 x + 2}\right)\, dx \]algebraSimplify the fraction by splitting it.✓ Proved
- \[ = - \frac{x^{2}}{4} - \frac{x}{2} + \left(\frac{x^{2}}{2} + x\right) \ln{\left(x + 1 \right)} + \int \frac{1}{2 x + 2}\, dx \]antiderivativeIntegrate the polynomial terms and handle the negative sign.✓ Proved
- \[ = - \frac{x^{2}}{4} - \frac{x}{2} + \left(\frac{x^{2}}{2} + x\right) \ln{\left(x + 1 \right)} + \frac{\ln{\left(x + 1 \right)}}{2} \]antiderivativeIntegrate the remaining logarithmic term.✓ Proved
- \[ = - \frac{x^{2}}{4} - \frac{x}{2} + \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \ln{\left(x + 1 \right)} \]simplifyCombine the log(x + 1) terms.✓ Proved
- \[ = - \frac{x^{2}}{4} - \frac{x}{2} + \frac{\left(x + 1\right)^{2} \ln{\left(x + 1 \right)}}{2} \]simplifyFactor the coefficient of the log term.✓ Proved
Answer \( - \frac{x^{2}}{4} - \frac{x}{2} + \left(\frac{x^{2}}{2} + x\right) \ln{\left(x + 1 \right)} + \frac{\ln{\left(x + 1 \right)}}{2} + C \)
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 undefined where x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 1 = 0 undefined where 2*x + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 undefined where x + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 1 = 0 undefined where 2*x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | lines differ by the constant -log(2)/2 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | lines differ by the constant log(2)/2 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 12 | ✓ 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 2 applies integration by parts but the antiderivative of (x+1) is (x+1)^2/2, not (x^2/2 + x). The missing +1/2 term means the subsequent algebraic manipulations are based on an incorrect expression.qwen3.6:27b-mlx: pass — The solution correctly applies integration by parts and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-08 — Step 2 applies integration by parts but the antiderivative of (x+1) is (x+1)^2/2, not (x^2/2 + x). The missing +1/2 term means the subsequent algebraic manipulations are based on an incorrect expression.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies integration by parts and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-08 — Step 2 incorrectly applies integration by parts: the antiderivative of (x+1) is (x+1)^2/2, not (x^2/2 + x). The missing +1/2 leads to an incorrect final result.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies integration by parts and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
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 2026-10-08 with SymPy 1.14.0.