Integral of \( \displaystyle \frac{3 x}{\left(3 x + 1\right) \left(3 x + 3\right)} \)
Problem 4.551 · medium
Find \( \displaystyle \int \frac{3 x}{\left(3 x + 1\right) \left(3 x + 3\right)} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{3 x}{\left(3 x + 1\right) \left(3 x + 3\right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(\frac{3}{6 x + 6} - \frac{1}{6 x + 2}\right)\, dx \]rewriteUse partial fraction decomposition.✓ Proved
- \[ = \int \left(- \frac{1}{6 x + 2}\right)\, dx + \int \frac{3}{6 x + 6}\, dx \]linearitySplit the integral into two parts.≈ Checked numerically
- \[ = - \frac{\int \frac{1}{3 x + 1}\, dx}{2} + \frac{3 \int \frac{1}{3 x + 3}\, dx}{2} \]algebraFactor out the constants.✓ Proved
- \[ = - \frac{\ln{\left(3 x + 1 \right)}}{6} + \frac{\ln{\left(3 x + 3 \right)}}{2} \]antiderivative simplifyIntegrate each term. Simplify the coefficients.✓ Proved
Answer \( - \frac{\ln{\left(x + \frac{1}{3} \right)}}{6} + \frac{\ln{\left(x + 1 \right)}}{2} + C \)
✓ Nihil obstat Lines: 5 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 + 3 = 0 undefined where 3*x + 1 = 0 undefined where 6*x + 6 = 0 undefined where 6*x + 2 = 0 |
| 3 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left -log(x + 1/3)/6 + log(3*x + 1)/6 - log(2)/3; numeric agreement only, at 24 of 24 sampled points undefined where 6*x + 6 = 0 undefined where 6*x + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | lines differ by the constant -log(3)/2 + log(2)/3 undefined where 6*x + 6 = 0 undefined where 6*x + 2 = 0 undefined where 3*x + 3 = 0 undefined where 3*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 3 = 0 undefined where 3*x + 1 = 0 log is undefined for non-positive arguments |
| 6 | ✓ 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)/6 - log(3*x + 1)/6 + log(3)/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: fail (style) — Step 2 is labeled 'rewrite' but the note explicitly states 'Use partial fraction decomposition'. The label should be 'partial-fractions' to accurately reflect the rule applied. Step 4 is labeled 'algebra' but performs the extraction of constant factors from the integral, which is better described as 'linearity' or 'constant-multiple' (though 'linearity' was already used in step 3, 'algebra' is too vague for a specific integral property; however, given the strict vocabulary, 'linearity' is the correct label for pulling constants out, making Step 4's label 'algebra' a misnomer for the operation performed, or Step 3 and 4 are redundant applications of linearity. The primary defect is the mismatch in Step 2's label vs note.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (style) 2026-10-07 — Step 2 is labeled 'rewrite' but the note explicitly states 'Use partial fraction decomposition'. The label should be 'partial-fractions' to accurately reflect the rule applied. Step 4 is labeled 'algebra' but performs the extraction of constant factors from the integral, which is better described as 'linearity' or 'constant-multiple' (though 'linearity' was already used in step 3, 'algebra' is too vague for a specific integral property; however, given the strict vocabulary, 'linearity' is the correct label for pulling constants out, making Step 4's label 'algebra' a misnomer for the operation performed, or Step 3 and 4 are redundant applications of linearity. The primary defect is the mismatch in Step 2's label vs note.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies partial fraction decomposition, linearity, and standard antiderivatives. The final expression is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.