Integral of \( \displaystyle \frac{1}{\left(3 x - 1\right)^{2} - 4} \)
Problem 4.1479 · medium Video
Find \( \displaystyle \int \frac{1}{\left(3 x - 1\right)^{2} - 4} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(3 x - 1\right)^{2} - 4}\, dx \]integralStart with the given integral.✓ Proved
- \[ = \int \frac{1}{\left(3 x - 3\right) \left(3 x + 1\right)}\, dx \]algebraFactor the denominator as a difference of squares.✓ Proved
- \[ = \int \frac{1}{\left(x + \frac{1}{3}\right) \left(9 x - 9\right)}\, dx \]algebraFactor out 3 from both linear terms.✓ Proved
- \[ = \int \left(- \frac{1}{12 x + 4} + \frac{1}{12 x - 12}\right)\, dx \]partial-fractionsApply partial fraction decomposition.✓ Proved
- \[ = \frac{\ln{\left(x - 1 \right)}}{12} - \frac{\ln{\left(x + \frac{1}{3} \right)}}{12} \]antiderivativeIntegrate each term using the rule for 1/x.✓ Proved
Answer \( \frac{\ln{\left(x - 1 \right)} - \ln{\left(x + \frac{1}{3} \right)}}{12} + C \)
✓ 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. 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)**2 - 4 = 0 undefined where 3*x + 1 = 0 undefined where 3*x - 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 undefined where 3*x - 3 = 0 undefined where x + 1/3 = 0 undefined where 9*x - 9 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1/3 = 0 undefined where 9*x - 9 = 0 undefined where 12*x - 12 = 0 undefined where 12*x + 4 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 12*x - 12 = 0 undefined where 12*x + 4 = 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: pass — The solution correctly applies algebraic manipulation, partial fraction decomposition, and integration rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-09 — The solution correctly applies algebraic manipulation, partial fraction decomposition, and integration rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-09qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly factors the denominator, applies partial fraction decomposition, and integrates term-by-term. Each step applies a single rule and is labeled correctly.gpt-oss:20b: pass 2026-11-08qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly factors the denominator, applies partial fraction decomposition, and integrates term-by-term. Each step applies exactly one rule from the allowed vocabulary and is labeled correctly.gpt-oss:20b: pass 2026-11-08
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 2026-11-09 with SymPy 1.14.0.