Integral of \( \displaystyle 3 x + 1 + \frac{1}{3 x + 1} \)
Problem 4.360 · medium
Find \( \displaystyle \int 3 x + 1 + \frac{1}{3 x + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \left(3 x + 1 + \frac{1}{3 x + 1}\right)\, dx \]integralStart with the integral of the given expression.✓ Proved
- \[ = \int 1\, dx + \int 3 x\, dx + \int \frac{1}{3 x + 1}\, dx \]linearitySplit the integral into three separate parts.✓ Proved
- \[ = \int 1\, dx + 3 \int x\, dx + \int \frac{1}{3 x + 1}\, dx \]linearityFactor out the constant 3 from the first term.✓ Proved
- \[ = \frac{3 x^{2}}{2} + x + \int \frac{1}{3 x + 1}\, dx \]antiderivativeIntegrate the first two terms.✓ Proved
- \[ = \frac{3 x^{2}}{2} + x + \frac{\ln{\left(3 x + 1 \right)}}{3} \]substitutionIntegrate the remaining term using substitution u = 3x + 1.✓ Proved
Answer \( \frac{3 x^{2}}{2} + x + \frac{\ln{\left(3 x + 1 \right)}}{3} + C \)
Every line of this solution was proved by the computer algebra system SymPy. 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 3*x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 log is undefined for non-positive arguments |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 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 4 applies the antiderivative rule to two separate terms simultaneously, violating the contract that each step must change only one thing. It should have integrated the first term in one step and the second term in a subsequent step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 4 applies the antiderivative rule to two separate terms simultaneously, violating the contract that each step must change only one thing. It should have integrated the first term in one step and the second term in a subsequent step.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 4 applies the antiderivative rule to two separate terms (3*x and 1) simultaneously, violating the contract that each step must change only one thing. Step 5 uses the label 'substitution' for a step that performs the final integration and algebraic simplification at once, rather than showing the substitution mechanics.gpt-oss:20b: pass 2026-10-05
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.