Integral of \( \displaystyle \frac{3 x + 4}{3 x + \left(3 x - 1\right)^{2} - 3} \)
Problem 4.1352 · hard Video
Find \( \displaystyle \int \frac{3 x + 4}{3 x + \left(3 x - 1\right)^{2} - 3} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{3 x + 4}{3 x + \left(3 x - 1\right)^{2} - 3}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{3 x + 4}{9 x^{2} - 3 x - 2}\, dx \]algebra simplifyExpand the squared term. Combine like terms in the denominator.✓ Proved
- \[ = \int \frac{3 x + 4}{\left(3 x - 2\right) \left(3 x + 1\right)}\, dx \]algebraFactor the quadratic denominator.✓ Proved
- \[ = \int \left(- \frac{1}{3 x + 1} + \frac{2}{3 x - 2}\right)\, dx \]partial-fractionsDecompose the integrand into partial fractions.✓ Proved
- \[ = \int \frac{2}{3 x - 2}\, dx - \int \frac{1}{3 x + 1}\, dx \]linearitySplit the integral into two parts.≈ Checked numerically
- \[ = \frac{2 \ln{\left(3 x - 2 \right)}}{3} - \frac{\ln{\left(3 x + 1 \right)}}{3} \]antiderivativeIntegrate each term using the substitution rule for log(ax+b).✓ Proved
Answer \( \frac{2 \ln{\left(x - \frac{2}{3} \right)} - \ln{\left(x + \frac{1}{3} \right)}}{3} + C \)
Lines: 6 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. How we verify
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*x - 1)**2 - 3 = 0 undefined where 9*x**2 - 3*x - 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 - 3*x - 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 - 3*x - 2 = 0 undefined where 3*x + 1 = 0 undefined where 3*x - 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 undefined where 3*x - 2 = 0 |
| 6 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2*log(x - 2/3)/3 - log(x + 1/3)/3 - 2*log(3*x - 2)/3 + log(3*x + 1)/3; numeric agreement only, at 24 of 24 sampled points undefined where 3*x + 1 = 0 undefined where 3*x - 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 undefined where 3*x - 2 = 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 - 2/3)/3 + log(x + 1/3)/3 + 2*log(3*x - 2)/3 - log(3*x + 1)/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: passqwen3.6:27b-mlx: fail (error) — Step 5 is algebraically incorrect; the partial fraction decomposition of (3x+4)/((3x-2)(3x+1)) is not 2/(3x-2) - 1/(3x+1). The correct decomposition is 1/(3x-2) + 2/(3x+1). Consequently, the final answer is wrong.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-27 — Step 5 is algebraically incorrect; the partial fraction decomposition of (3x+4)/((3x-2)(3x+1)) is not 2/(3x-2) - 1/(3x+1). The correct decomposition is 1/(3x-2) + 2/(3x+1). Consequently, the final answer is wrong.gpt-oss:20b: pass 2026-10-27qwen3.6:27b-mlx: pass 2026-10-26 — The solution correctly expands, simplifies, factors, and decomposes the integrand. The final integration step correctly applies the antiderivative rule for rational functions, and the result matches the stated answer.gpt-oss:20b: pass 2026-10-26qwen3.6:27b-mlx: fail (error) 2026-10-26 — Step 5 is mathematically incorrect; the partial fraction decomposition of (3x+4)/((3x-2)(3x+1)) is 2/(3x-2) + 1/(3x+1), not 2/(3x-2) - 1/(3x+1). Consequently, the final answer has the wrong sign for the second term.gpt-oss:20b: pass 2026-10-26
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 2026-10-27 with SymPy 1.14.0.