Integral of \( \displaystyle \frac{6 x - 1}{\left(3 x - 1\right)^{2} - 1} \)
Problem 4.1481 · medium Video
Find \( \displaystyle \int \frac{6 x - 1}{\left(3 x - 1\right)^{2} - 1} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{6 x - 1}{\left(3 x - 1\right)^{2} - 1}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{6 x - 1}{9 x^{2} - 6 x}\, dx \]algebraExpand the denominator.✓ Proved
- \[ = \int \frac{6 x - 1}{3 x \left(3 x - 2\right)}\, dx \]simplifyFactor the denominator.✓ Proved
- \[ = \int \left(\frac{3}{6 x - 4} + \frac{1}{6 x}\right)\, dx \]partial-fractionsPerform partial fraction decomposition.✓ Proved
- \[ = \int \frac{1}{6 x}\, dx + \int \frac{3}{6 x - 4}\, dx \]linearitySplit the integral into two parts.≈ Checked numerically
- \[ = \frac{\int \frac{1}{x}\, dx}{6} + \frac{3 \int \frac{1}{3 x - 2}\, dx}{2} \]rewriteFactor out the constants from the denominators.✓ Proved
- \[ = \frac{\ln{\left(x \right)}}{6} + \frac{\ln{\left(3 x - 2 \right)}}{2} \]antiderivativeEvaluate the integrals.✓ Proved
Answer \( \frac{\ln{\left(x \right)}}{6} + \frac{\ln{\left(x - \frac{2}{3} \right)}}{2} + C \)
Lines: 6 proved, 2 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 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x - 1)**2 - 1 = 0 undefined where 9*x**2 - 6*x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 - 6*x = 0 undefined where x = 0 undefined where 3*x - 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 3*x - 2 = 0 undefined where 6*x - 4 = 0 |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log(x - 2/3)/2 - log(6*x - 4)/2; numeric agreement only, at 24 of 24 sampled points undefined where x = 0 undefined where 6*x - 4 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | lines differ by the constant log(2)/2 undefined where x = 0 undefined where 6*x - 4 = 0 undefined where 3*x - 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 3*x - 2 = 0 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 -log(x - 2/3)/2 + log(3*x - 2)/2; 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: passqwen3.6:27b-mlx: pass — The solution correctly applies partial fraction decomposition and integration rules. The final answer matches the stated answer, noting that log(3x-2) and log(x-2/3) differ only by a constant (log 3), which is absorbed into the omitted constant of integration.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-10 — The solution correctly applies partial fraction decomposition and integration rules. The final answer matches the stated answer, noting that log(3x-2) and log(x-2/3) differ only by a constant (log 3), which is absorbed into the omitted constant of integration.gpt-oss:20b: pass 2026-11-10qwen3.6:27b-mlx: fail (error) 2026-11-09 — Step 6 claims to use 'rewrite' to factor out constants, but it actually performs a u-substitution (or change of variables) for the second term, which is a distinct operation from simple algebraic rewriting. Specifically, Integral(1/(3x-2), dx) becomes (1/3)Integral(1/u, du), which is not captured by the label 'rewrite'. Furthermore, the label 'rewrite' is not in the allowed vocabulary for this specific transformation; 'substitution' would be the correct label if the step were split, or the step is too complex for a single label if it combines factoring and substitution logic incorrectly. Actually, looking closer: Step 6 goes from Integral(3/(2*(3*x - 2)), x) to 3/2 * Integral(1/(3*x - 2), x). This is just pulling out the constant 3/2. That is 'linearity' or 'algebra'. But the note says 'Factor out the constants from the denominators'. The step itself is just pulling 3/2 out. That is fine. Wait, let's look at Step 7. Step 7 goes from 1/6 * Integral(1/x, x) + 3/2 * Integral(1/(3*x - 2), x) to 1/6 * log(x) + 1/2 * log(3*x - 2). The integral of 1/(3x-2) is (1/3)log(3x-2). So 3/2 * (1/3)log(3x-2) = 1/2 log(3x-2). The math is correct. The label is 'antiderivative'. This step applies the antiderivative rule to both terms. Is that one rule? Yes, 'antiderivative' covers finding the primitive. Let's re-read the contract. 'Each step changes ONE thing.' Step 6 changes the coefficient of the second integral. Step 7 changes both integrals to logs. Step 7 applies the antiderivative rule to two separate terms. Is that allowed? 'A step that applies one rule to one term, leaving the rest untouched, is fine.' Step 7 applies the antiderivative rule to *both* terms simultaneously. This violates the 'ONE thing' rule. It should be split into two steps: one for the first term, one for the second. Or, if 'antiderivative' is considered a global operation on the expression, it might be acceptable, but usually, linearity splits them, and then you integrate each. Step 5 split them. Step 6 pulled constants. Step 7 integrated both. Integrating two separate terms at once is applying the rule twice. This is a defect.gpt-oss:20b: pass 2026-11-09qwen3.6:27b-mlx: pass 2026-11-09 — The solution correctly applies algebraic manipulation, partial fraction decomposition, and standard integration rules. Each step changes only one aspect of the expression and uses valid labels from the allowed vocabulary.gpt-oss:20b: pass 2026-11-09
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 2026-11-10 with SymPy 1.14.0.