Integral of \( \displaystyle \frac{1}{\left(2 x - 2\right) \left(2 x + 1\right)} \)
Problem 4.1281 · medium Video
Find \( \displaystyle \int \frac{1}{\left(2 x - 2\right) \left(2 x + 1\right)} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(2 x - 2\right) \left(2 x + 1\right)}\, dx \]integral algebraStart with the integral of the given function. Factor out the constant from the first term.✓ Proved
- \[ = \int \left(- \frac{1}{6 x + 3} + \frac{1}{6 x - 6}\right)\, dx \]partial-fractionsUse partial fraction decomposition.✓ Proved
- \[ = \int \frac{1}{6 x - 6}\, dx - \int \frac{1}{6 x + 3}\, dx \]linearitySplit the integral into two parts.≈ Checked numerically
- \[ = \frac{\int \frac{1}{2 x - 2}\, dx}{3} - \frac{\int \frac{1}{2 x + 1}\, dx}{3} \]linearityFactor out the constant 1/3 from both integrals.✓ Proved
- \[ = \frac{\ln{\left(2 x - 2 \right)}}{6} - \frac{\ln{\left(2 x + 1 \right)}}{6} \]antiderivative simplifyApply the antiderivative rule for 1/(ax+b). Simplify the coefficients.✓ Proved
Answer \( \frac{\ln{\left(x - 1 \right)} - \ln{\left(x + \frac{1}{2} \right)}}{6} + C \)
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 2*x - 2 = 0 undefined where 2*x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 2 = 0 undefined where 2*x + 1 = 0 undefined where 6*x - 6 = 0 undefined where 6*x + 3 = 0 |
| 4 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left -log(x + 1/2)/6 + log(2*x + 1)/6 - log(6)/6 + log(3)/6; numeric agreement only, at 24 of 24 sampled points undefined where 6*x - 6 = 0 undefined where 6*x + 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 6*x - 6 = 0 undefined where 6*x + 3 = 0 undefined where 2*x - 2 = 0 undefined where 2*x + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 2 = 0 undefined where 2*x + 1 = 0 log is undefined for non-positive arguments |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 log(x + 1/2)/6 - log(2*x + 1)/6 + log(2)/6; 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 3 is mathematically incorrect; the partial fraction decomposition of 1/((2x-2)(2x+1)) is 1/3 * (1/(2x+1) - 1/(2x-2)), but the step writes 1/(3(2x-2)) - 1/(3(2x+1)), which has the wrong signs and incorrect denominators (it should be 1/6 * (1/(x-1) - 1/(x+1/2)) or equivalent). Consequently, the final answer has the wrong sign for the second term and incorrect arguments inside the logarithms compared to the standard form derived from the correct decomposition.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 3 is mathematically incorrect; the partial fraction decomposition of 1/((2x-2)(2x+1)) is 1/3 * (1/(2x+1) - 1/(2x-2)), but the step writes 1/(3(2x-2)) - 1/(3(2x+1)), which has the wrong signs and incorrect denominators (it should be 1/6 * (1/(x-1) - 1/(x+1/2)) or equivalent). Consequently, the final answer has the wrong sign for the second term and incorrect arguments inside the logarithms compared to the standard form derived from the correct decomposition.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies partial fraction decomposition, linearity, and standard antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies partial fraction decomposition, linearity, and standard antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-18
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 2026-10-19 with SymPy 1.14.0.