Integral of \( \displaystyle \frac{1}{\left(3 x - 2\right) \left(3 x + 1\right)} \)
Problem 4.372 · medium
Find \( \displaystyle \int \frac{1}{\left(3 x - 2\right) \left(3 x + 1\right)} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(3 x - 2\right) \left(3 x + 1\right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(- \frac{1}{3 \left(3 x + 1\right)} + \frac{1}{3 \left(3 x - 2\right)}\right)\, dx \]partial-fractionsDecompose the integrand using partial fraction decomposition.✓ Proved
- \[ = \frac{\int \frac{1}{3 x - 2}\, dx}{3} - \frac{\int \frac{1}{3 x + 1}\, dx}{3} \]linearityApply linearity to split the integral into two parts.≈ Checked numerically
- \[ = \frac{\ln{\left(3 x - 2 \right)}}{9} - \frac{\ln{\left(3 x + 1 \right)}}{9} \]antiderivative algebraIntegrate each term using the substitution rule for log(ax+b). Simplify the coefficients.✓ Proved
Answer \( \frac{\ln{\left(x - \frac{2}{3} \right)} - \ln{\left(x + \frac{1}{3} \right)}}{9} + C \)
Lines: 4 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.
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 - 2 = 0 undefined where 3*x + 1 = 0 |
| 3 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log(x - 2/3)/9 - log(x + 1/3)/9 - log(3*x - 2)/9 + log(3*x + 1)/9; numeric agreement only, at 24 of 24 sampled points undefined where 3*x - 2 = 0 undefined where 3*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 2 = 0 undefined where 3*x + 1 = 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 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -log(x - 2/3)/9 + log(x + 1/3)/9 + log(3*x - 2)/9 - log(3*x + 1)/9; 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 fractions, linearity, and the antiderivative rule for logarithmic forms. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies partial fractions, linearity, and the antiderivative rule for logarithmic forms. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: fail (misleading) 2026-10-05 — The solution omits the absolute‑value bars in the logarithms, which is a common source of error when integrating 1/(ax+b).
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 2026-10-05 with SymPy 1.14.0.