Integral of \( \displaystyle \frac{1}{\left(3 x - 1\right)^{2}} \)
Problem 4.1405 · medium Video
Find \( \displaystyle \int \frac{1}{\left(3 x - 1\right)^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(3 x - 1\right)^{2}}\, dx \]integral algebra algebra algebra rewrite rewriteStart with the integral of the given function. Rewrite the integrand by multiplying and dividing by 3. Rearrange the terms for clarity. Simplify the constant factor 3 * (1/3) to 1. Rewrite the power using the exponential and logarithm. Move the exponent inside the exponential function.✓ Proved
- \[ = \int \frac{1}{9 \left(x - \frac{1}{3}\right)^{2}}\, dx \]algebra algebraExpand the logarithm. Distribute the -2.✓ Proved
- \[ = \frac{\int \frac{1}{\left(x - \frac{1}{3}\right)^{2}}\, dx}{9} \]linearity rewritePull out the constant factor. Rewrite the exponential term back to power form.✓ Proved
- \[ = - \frac{1}{9 \left(x - \frac{1}{3}\right)} \]antiderivative simplify simplifyIntegrate the power function. Simplify the constant factor. Simplify the expression.✓ Proved
- \[ = - \frac{1}{9 x - 3} \]simplify simplify simplifyDistribute the 1/9 into the denominator. Factor the denominator. Expand the denominator.✓ Proved
Answer \( - \frac{1}{9 x - 3} + C \)
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. 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 - 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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0 undefined where x - 1/3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1/3 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1/3 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1/3 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1/3 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1/3 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1/3 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1/3 = 0 undefined where 9*x - 3 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x - 3 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x - 3 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 9*x - 3 = 0 |
| 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: pass — The solution is mathematically correct and adheres to the one-change-per-step constraint. The steps logically transform the integrand, apply the power rule for integration, and simplify the result to the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-01 — The solution is mathematically correct and adheres to the one-change-per-step constraint. The steps logically transform the integrand, apply the power rule for integration, and simplify the result to the stated answer.gpt-oss:20b: pass 2026-11-01qwen3.6:27b-mlx: inconclusive 2026-10-31 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 7 applies the logarithmic identity log(a*b) = log(a) + log(b) to log(3*x - 1), treating it as log(3 * (x - 1/3)). While algebraically true forgpt-oss:20b: pass 2026-10-31qwen3.6:27b-mlx: fail (style) 2026-10-31 — [domain objection, downgraded to style] Step 7 incorrectly applies the logarithm product rule log(3x-1) = log(3) + log(x-1/3), which is only valid when 3x-1 > 0. Since the domain includes regions where 3x-1 < 0, this step is mathematically invalid without absolute values or complex logarithms, leading to a potential sign error in the constant factor for those regions.gpt-oss:20b: pass 2026-10-31
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 2026-11-01 with SymPy 1.14.0.