Integral of \( \displaystyle \frac{3 x + 5}{9 x^{2} + 3 x - 2} \)
Problem 4.558 · medium
Find \( \displaystyle \int \frac{3 x + 5}{9 x^{2} + 3 x - 2} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{3 x + 5}{9 x^{2} + 3 x - 2}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{3 x + 5}{\left(3 x - 1\right) \left(3 x + 2\right)}\, dx \]algebraFactor the denominator.✓ Proved
- \[ = \int \left(- \frac{1}{3 x + 2} + \frac{2}{3 x - 1}\right)\, dx \]partial-fractionsDecompose the integrand into partial fractions.✓ Proved
- \[ = \int \frac{2}{3 x - 1}\, dx - \int \frac{1}{3 x + 2}\, dx \]linearitySplit the integral into two parts.≈ Checked numerically
- \[ = \frac{2 \int \frac{1}{x - \frac{1}{3}}\, dx}{3} - \frac{\int \frac{1}{x + \frac{2}{3}}\, dx}{3} \]rewriteRewrite the denominators to isolate x.✓ Proved
- \[ = \frac{2 \ln{\left(3 x - 1 \right)}}{3} - \frac{\ln{\left(3 x + 2 \right)}}{3} \]antiderivativeIntegrate the terms using the rule for 1/(ax+b).✓ Proved
Answer \( \frac{2 \ln{\left(x - \frac{1}{3} \right)} - \ln{\left(x + \frac{2}{3} \right)}}{3} + C \)
Lines: 5 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 9*x**2 + 3*x - 2 = 0 undefined where 3*x + 2 = 0 undefined where 3*x - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 2 = 0 undefined where 3*x - 1 = 0 |
| 4 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2*log(x - 1/3)/3 - log(x + 2/3)/3 - 2*log(3*x - 1)/3 + log(3*x + 2)/3; numeric agreement only, at 24 of 24 sampled points undefined where 3*x + 2 = 0 undefined where 3*x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 2 = 0 undefined where 3*x - 1 = 0 undefined where x + 2/3 = 0 undefined where x - 1/3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 2/3 = 0 undefined where x - 1/3 = 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 -2*log(x - 1/3)/3 + log(x + 2/3)/3 + 2*log(3*x - 1)/3 - log(3*x + 2)/3; 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: fail (error) — Step 6 applies the antiderivative rule incorrectly: it integrates 1/(x‑1/3) as log(3*x‑1) instead of log|x‑1/3| (or equivalently (1/3)log|3*x‑1|). This mixes up the chain rule and omits the necessary absolute value, producing an incorrect antiderivative.qwen3.6:27b-mlx: fail (error) — Step 6 is mathematically incorrect. The antiderivative of 1/(x - 1/3) is log(x - 1/3), not log(3*x - 1). The step claims to integrate the terms from Step 5 (which have denominators x - 1/3 and x + 2/3) but produces results corresponding to the denominators in Step 3 (3*x - 1 and 3*x + 2) without accounting for the chain rule factor of 3, resulting in an incorrect coefficient and argument.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — Step 6 applies the antiderivative rule incorrectly: it integrates 1/(x‑1/3) as log(3*x‑1) instead of log|x‑1/3| (or equivalently (1/3)log|3*x‑1|). This mixes up the chain rule and omits the necessary absolute value, producing an incorrect antiderivative.qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 6 is mathematically incorrect. The antiderivative of 1/(x - 1/3) is log(x - 1/3), not log(3*x - 1). The step claims to integrate the terms from Step 5 (which have denominators x - 1/3 and x + 2/3) but produces results corresponding to the denominators in Step 3 (3*x - 1 and 3*x + 2) without accounting for the chain rule factor of 3, resulting in an incorrect coefficient and argument.qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 6 is mathematically incorrect: the antiderivative of 1/(x - a) is log(x - a), not log(3x - 1). The step fails to account for the chain rule facgpt-oss:20b: fail (error) 2026-10-07 — Step 6 integrates 2/(3*x-1) and 1/(3*x+2) incorrectly. The antiderivative of 1/(3*x-1) is (1/3)ln|3*x-1|, not ln(3*x-1). Likewise, 1/(3*x+2) integrates to (1/3)ln|3*x+2|. The solution should use ln|x-1/3| and ln|x+2/3| or equivalently (1/3)ln|3*x-1| and (1/3)ln|3*x+2|, preserving the absolute value. The current step applies the wrong rule and omits the absolute value, leading to an incorrect antiderivative.
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.