Integral of \( \displaystyle \frac{1}{\left(3 x - 1\right) \left(3 x + 2\right)} \)
Problem 4.462 · medium
Find \( \displaystyle \int \frac{1}{\left(3 x - 1\right) \left(3 x + 2\right)} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(3 x - 1\right) \left(3 x + 2\right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(- \frac{1}{3 \left(3 x + 2\right)} + \frac{1}{3 \left(3 x - 1\right)}\right)\, dx \]partial-fractionsDecompose the integrand using partial fraction decomposition.✓ Proved
- \[ = \frac{\int \frac{1}{3 x - 1}\, dx}{3} - \frac{\int \frac{1}{3 x + 2}\, dx}{3} \]linearityDistribute the constant and split the integral into two parts.≈ Checked numerically
- \[ = \frac{\ln{\left(3 x - 1 \right)}}{9} - \frac{\ln{\left(3 x + 2 \right)}}{9} \]antiderivative algebraIntegrate each term using the rule for 1/(ax+b). Simplify the coefficients.✓ Proved
Answer \( \frac{\ln{\left(x - \frac{1}{3} \right)} - \ln{\left(x + \frac{2}{3} \right)}}{9} + C \)
✓ Nihil obstat Lines: 4 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 - 1/3)/9 - log(x + 2/3)/9 - log(3*x - 1)/9 + log(3*x + 2)/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 - 1/3)/9 + log(x + 2/3)/9 + log(3*x - 1)/9 - log(3*x + 2)/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 fraction decomposition, linearity, and the standard antiderivative rule for 1/(ax+b). 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: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies partial fraction decomposition, linearity, and the standard antiderivative rule for 1/(ax+b). Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies partial fraction decomposition, linearity, and the standard antiderivative rule for 1/(ax+b). The final answer matches the stated answer via algebraic equivalence.gpt-oss:20b: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.