Integral of \( \displaystyle 3 x + \frac{1}{3 x} \)
Problem 4.556 · medium
Find \( \displaystyle \int 3 x + \frac{1}{3 x} \, dx \). (Omit the constant of integration.)
- \[ \int \left(3 x + \frac{1}{3 x}\right)\, dx \]integralStart with the integral of the given expression.✓ Proved
- \[ = \int \frac{1}{3 x}\, dx + \int 3 x\, dx \]linearitySplit the integral into two parts using linearity.✓ Proved
- \[ = \frac{\int \frac{1}{x}\, dx}{3} + \int 3 x\, dx \]linearityFactor out the constant 1/3 from the second integral.✓ Proved
- \[ = \frac{3 x^{2}}{2} + \frac{\ln{\left(x \right)}}{3} \]antiderivativeEvaluate the integrals using the power rule and the natural logarithm rule.✓ Proved
Answer \( \frac{3 x^{2}}{2} + \frac{\ln{\left(x \right)}}{3} + C \)
✓ Nihil obstat 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. 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 x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 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: fail (style) — [domain objection, downgraded to style] The antiderivative of 1/x is ln|x|, not ln(x). The solution omits the absolute value, which could mislead a student about the domain of the logarithm.qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-07 — [domain objection, downgraded to style] The antiderivative of 1/x is ln|x|, not ln(x). The solution omits the absolute value, which could mislead a student about the domain of the logarithm.qwen3.6:27b-mlx: pass 2026-10-07gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: 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.