Integral of \( \displaystyle \frac{1}{3 x \left(3 x + 1\right)^{2}} \)
Problem 4.368 · medium
Find \( \displaystyle \int \frac{1}{3 x \left(3 x + 1\right)^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{3 x \left(3 x + 1\right)^{2}}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(- \frac{1}{3 x + 1} - \frac{1}{\left(3 x + 1\right)^{2}} + \frac{1}{3 x}\right)\, dx \]rewriteDecompose the integrand using partial fraction decomposition.✓ Proved
- \[ = \int \frac{1}{3 x}\, dx - \int \frac{1}{\left(3 x + 1\right)^{2}}\, dx - \int \frac{1}{3 x + 1}\, dx \]linearityDistribute the integral across the terms.≈ Checked numerically
- \[ = \frac{\ln{\left(x \right)}}{3} - \int \frac{1}{\left(3 x + 1\right)^{2}}\, dx - \int \frac{1}{3 x + 1}\, dx \]antiderivativeIntegrate the first term.✓ Proved
- \[ = \frac{\ln{\left(x \right)}}{3} - \frac{\ln{\left(3 x + 1 \right)}}{3} - \int \frac{1}{\left(3 x + 1\right)^{2}}\, dx \]antiderivativeIntegrate the second term.✓ Proved
- \[ = \frac{\ln{\left(x \right)}}{3} - \frac{\ln{\left(3 x + 1 \right)}}{3} + \frac{1}{9 x + 3} \]antiderivativeIntegrate the final term using the power rule.✓ Proved
Answer \( \frac{\ln{\left(x \right)}}{3} - \frac{\ln{\left(x + \frac{1}{3} \right)}}{3} + \frac{1}{9 x + 3} + C \)
Lines: 5 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 x = 0 undefined where 3*x + 1 = 0 |
| 3 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left -log(x + 1/3)/3 + log(3*x + 1)/3; numeric agreement only, at 24 of 24 sampled points undefined where x = 0 undefined where 3*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 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 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 undefined where 9*x + 3 = 0 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left log(x + 1/3)/3 - log(3*x + 1)/3; numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where 9*x + 3 = 0 |
| 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies partial fraction decomposition, linearity, and antiderivative rules in separate steps. The final answer is algebraically equivalent to the stated answer.gpt-oss:20b: fail (error) 2026-10-05 — Step 6 applies the antiderivative of (3x+1)^{-2} incorrectly: ∫(3x+1)^{-2}dx = -(1/3)(3x+1)^{-1}, not +(1/3)(3x+1)^{-1}. The sign of the final term is wrong.
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.