Integral of \( \displaystyle \ln{\left(3 x - 1 \right)} \)
Problem 4.1262 · easy Video
Find \( \displaystyle \int \ln{\left(3 x - 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \ln{\left(3 x - 1 \right)}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = - \int \frac{3 x}{3 x - 1}\, dx + \int \left(\frac{3 x}{3 x - 1} + \ln{\left(3 x - 1 \right)}\right)\, dx \]partsUse integration by parts with u = log(3*x - 1) and dv = dx.✓ Proved
- \[ = x \ln{\left(3 x - 1 \right)} - \int \frac{3 x}{3 x - 1}\, dx \]simplify algebraSimplify the integration by parts formula. Rewrite the numerator to facilitate division.✓ Proved
- \[ = x \ln{\left(3 x - 1 \right)} - \int \left(1 + \frac{1}{3 x - 1}\right)\, dx \]algebraSplit the fraction into two parts.✓ Proved
- \[ = x \ln{\left(3 x - 1 \right)} - \int 1\, dx - \int \frac{1}{3 x - 1}\, dx \]linearityDistribute the integral sign.✓ Proved
- \[ = x \ln{\left(3 x - 1 \right)} - x - \int \frac{1}{3 x - 1}\, dx \]simplifyEvaluate the integral of 1.✓ Proved
- \[ = x \ln{\left(3 x - 1 \right)} - x - \frac{\ln{\left(3 x - 1 \right)}}{3} \]antiderivativeEvaluate the remaining integral using substitution.✓ Proved
- \[ = - x + \left(x - \frac{1}{3}\right) \ln{\left(3 x - 1 \right)} \]simplifyCombine the logarithmic terms.✓ Proved
Answer \( x \ln{\left(3 x - 1 \right)} - x - \frac{\ln{\left(3 x - 1 \right)}}{3} + C \)
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 undefined where 3*x - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 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: 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: passqwen3.6:27b-mlx: pass — The solution correctly applies integration by parts, algebraic manipulation, and standard antiderivatives. Each step changes only one aspect of the expression 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, algebraic manipulation, and standard antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies integration by parts, algebraic manipulation, and standard antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies integration by parts, algebraic manipulation, and standard antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-17 — Step 2 incorrectly rewrites the integral: it adds an extra integral of log(3*x-1) that should not be present. The correct integration by parts yields x*log(3*x-1) - ∫(3*x/(3*x-1))dx, not the expression shown.
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.