Integral of \( \displaystyle \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} \)
Problem 4.1284 · medium Video
Find \( \displaystyle \int \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(3 x - 1\right) \ln{\left(3 x - 1 \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(3 x - 1\right) \left(\ln{\left(x - \frac{1}{3} \right)} + \ln{\left(3 \right)}\right)\, dx \]rewriteUse the property log(ab) = log(a) + log(b).≈ Checked numerically
- \[ = \int \left(3 x - 1\right) \ln{\left(3 \right)}\, dx + \int \left(3 x - 1\right) \ln{\left(x - \frac{1}{3} \right)}\, dx \]linearityDistribute the term (3*x - 1).✓ Proved
- \[ = \int \left(3 x - 1\right) \ln{\left(x - \frac{1}{3} \right)}\, dx + \ln{\left(3 \right)} \int \left(3 x - 1\right)\, dx \]linearityPull out the constant log(3).✓ Proved
- \[ = \left(\frac{3 x^{2}}{2} - x\right) \ln{\left(3 \right)} + \int \left(3 x - 1\right) \ln{\left(x - \frac{1}{3} \right)}\, dx \]antiderivative algebraIntegrate the polynomial part. Rewrite 3*x - 1 as 3*(x - 1/3).✓ Proved
- \[ = \left(\frac{3 x^{2}}{2} - x\right) \ln{\left(3 \right)} + 3 \int \left(x - \frac{1}{3}\right) \ln{\left(x - \frac{1}{3} \right)}\, dx \]linearityPull out the constant 3.✓ Proved
- \[ = \left(\frac{3 x^{2}}{2} - x\right) \ln{\left(3 \right)} + 3 \int x \ln{\left(x - \frac{1}{3} \right)}\, dx - 3 \int \frac{\ln{\left(x - \frac{1}{3} \right)}}{3}\, dx \]linearitySplit the integrand into two parts.✓ Proved
- \[ = \left(\frac{3 x^{2}}{2} - x\right) \ln{\left(3 \right)} + 3 \int x \ln{\left(x - \frac{1}{3} \right)}\, dx - \int \ln{\left(x - \frac{1}{3} \right)}\, dx \]linearity algebraDistribute the 3 and simplify the constant term. Rewrite x as (x - 1/3) + 1/3.✓ Proved
- \[ = \left(\frac{3 x^{2}}{2} - x\right) \ln{\left(3 \right)} + 3 \int \left(x - \frac{1}{3}\right) \ln{\left(x - \frac{1}{3} \right)}\, dx + 3 \int \frac{\ln{\left(x - \frac{1}{3} \right)}}{3}\, dx - \int \ln{\left(x - \frac{1}{3} \right)}\, dx \]linearityDistribute the log(x - 1/3) inside the integral.✓ Proved
- \[ = \left(\frac{3 x^{2}}{2} - x\right) \ln{\left(3 \right)} + 3 \int \left(x - \frac{1}{3}\right) \ln{\left(x - \frac{1}{3} \right)}\, dx \]algebra simplify simplifySimplify the constant coefficient of the second integral. The two integrals of log(x - 1/3) cancel out. Clean up the expression.✓ Proved
- \[ = \frac{3 \left(x - \frac{1}{3}\right)^{2} \ln{\left(x - \frac{1}{3} \right)}}{2} + \left(\frac{3 x^{2}}{2} - x\right) \ln{\left(3 \right)} - 3 \int \left(\frac{x}{2} - \frac{1}{6}\right)\, dx \]parts algebraApply integration by parts to the integral term with u = log(x - 1/3) and dv = (x - 1/3)dx. Simplify the integrand in the second part.≈ Checked numerically
- \[ = \frac{3 \left(x - \frac{1}{3}\right)^{2} \ln{\left(x - \frac{1}{3} \right)}}{2} + \left(\frac{3 x^{2}}{2} - x\right) \ln{\left(3 \right)} - \frac{3 \int \left(x - \frac{1}{3}\right)\, dx}{2} \]algebraDistribute the 3 into the parentheses.✓ Proved
- \[ = - \frac{3 x^{2}}{4} + \frac{x}{2} + \frac{3 \left(x - \frac{1}{3}\right)^{2} \ln{\left(x - \frac{1}{3} \right)}}{2} + \left(\frac{3 x^{2}}{2} - x\right) \ln{\left(3 \right)} \]antiderivative algebraIntegrate the remaining polynomial term. Distribute the 3/2 and expand the polynomial.✓ Proved
- \[ = \frac{3 \left(x - \frac{1}{3}\right)^{2} \ln{\left(x - \frac{1}{3} \right)}}{2} - \frac{3 \left(x - \frac{1}{3}\right)^{2}}{4} + \left(\frac{3 x^{2}}{2} - x\right) \ln{\left(3 \right)} \]simplifyCombine the polynomial terms into a more compact form.✓ Proved
Answer \( - \frac{3 x^{2}}{4} + \frac{x}{2} + \left(\frac{3 x^{2}}{2} - x\right) \ln{\left(3 x - 1 \right)} + \frac{\ln{\left(3 x - 1 \right)}}{6} + C \)
Lines: 18 proved, 3 checked numerically. 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left x*(3*x*(1 - log(9)) - 3*x - 2*(3*x - 2)*log(x - 1/3) + 2*(3*x - 2)*log(3*x - 1) + log(81))/4; numeric agreement only, at 24 of 24 sampled points 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 | simplify(a - b) reduced to 0 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 |
| 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 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 15 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left -log(x - 1/3)/6 + log(3*x - 1)/6; numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 19 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 20 | ✓ Proved | sympy 1.14.0 | lines differ by the constant 1/12 log is undefined for non-positive arguments |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left 3*x**2/4 - x*(3*x - 2)*log(3*x - 1)/2 + x*(3*x - 2)*log(3)/2 - x/2 + (3*x - 1)**2*log(x - 1/3)/6 - (3*x - 1)**2/12 - log(3*x - 1)/6; numeric agreement only, at 24 of 24 sampled points 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 20 combines the polynomial terms incorrectly: -3/4*(x-1/3)^2 expands to -3/4*x^2 + 3/2*x - 1/12, which does not match the earlier result -3/4*x^2 + x/2. The constant term and the coefficient of x are wrong, so the final expression is not algebraically equivalent to the previous step.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 2 applies the identity log(3x-1) = log(3) + log(x-1/3), which is invalid for negative arguments of the original log function (where 3x-1
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-20 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 2 applies the identity log(3x-1) = log(3) + log(x-1/3), which is invalid for negative arguments of the original log function (where 3x-1gpt-oss:20b: fail (error) 2026-10-20 — Step 20 combines the polynomial terms incorrectly: -3/4*(x-1/3)^2 expands to -3/4*x^2 + 3/2*x - 1/12, which does not match the earlier result -3/4*x^2 + x/2. The constant term and the coefficient of x are wrong, so the final expression is not algebraically equivalent to the previous step.qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 2 incorrectly applies the logarithm property log(ab) = log(a) + log(b) to log(3x - 1), which is not a product of independent terms but a linear expression. This is a fundamental mathematical error that invalidates the subsequent steps.gpt-oss:20b: fail (style) 2026-10-19 — Step 3 applies both algebraic expansion and linearity of the integral in one move, and step 8 similarly combines an algebraic split with linearity. Each step should modify only one element using a single named rule.qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 2 incorrectly applies the logarithm property log(ab) = log(a) + log(b) to log(3x-1), which is not a product of independent terms but a composite function; log(3x-1) cannot be split into log(3) + log(x-1/3) without absolute values and careful handling of the argument's sign, but more importantly, the algebraic manipulation in subsequent steps (specifically steps 10-13) creates a circular loop that cancels out the term being integrated, leading to an incorrect final result that does not match the stated answer or the correct antiderivative.gpt-oss:20b: fail (error) 2026-10-19 — Step 20 incorrectly rewrites the polynomial part. The expression “-3/4*x**2 + x/2” is not equivalent to “-3/4*(x-1/3)**2”; the latter adds an extra constant term “-1/12”. This introduces an error in the final antiderivative.
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 2026-10-20 with SymPy 1.14.0.