Integral of \( \displaystyle \frac{1}{\left(2 x - 1\right) \left(2 x + 2\right)} \)
Problem 4.553 · medium
Find \( \displaystyle \int \frac{1}{\left(2 x - 1\right) \left(2 x + 2\right)} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(2 x - 1\right) \left(2 x + 2\right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{1}{4 x^{2} + 2 x - 2}\, dx \]algebraExpand the denominator.✓ Proved
- \[ = \int \frac{1}{\left(x + 1\right) \left(4 x - 2\right)}\, dx \]algebraFactor out the constant 2 from the denominator.✓ Proved
- \[ = \int \left(- \frac{1}{6 x + 6} + \frac{1}{6 x - 3}\right)\, dx \]partial-fractionsPerform partial fraction decomposition.✓ Proved
- \[ = \int \frac{1}{6 x - 3}\, dx - \int \frac{1}{6 x + 6}\, dx \]linearitySplit the integral into two parts.≈ Checked numerically
- \[ = \frac{\ln{\left(2 x - 1 \right)}}{6} - \frac{\ln{\left(2 x + 2 \right)}}{6} \]antiderivative simplifyIntegrate each term using the substitution u = ax + b. Factor out the common term.✓ Proved
Answer \( \frac{\ln{\left(x - \frac{1}{2} \right)} - \ln{\left(x + 1 \right)}}{6} + C \)
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.
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 undefined where 4*x**2 + 2*x - 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 2*x - 2 = 0 undefined where 4*x - 2 = 0 undefined where x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x - 2 = 0 undefined where x + 1 = 0 undefined where 6*x + 6 = 0 undefined where 6*x - 3 = 0 |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log(x - 1/2)/6 - log(2*x - 1)/6 - log(3)/6 + log(6)/6; numeric agreement only, at 24 of 24 sampled points undefined where 6*x + 6 = 0 undefined where 6*x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 6*x + 6 = 0 undefined where 6*x - 3 = 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 4 is mathematically incorrect. The partial fraction decomposition of 1/((2x-1)(2x+2)) is 1/6 * (1/(2x-1) - 1/(2x+2)), which simplifies to 1/(6(2x-1)) - 1/(6(2x+2)). The step claims the coefficients are 1/3 and -1/3, which is off by a factor of 2. Consequently, the final answer is incorrect (it should be divided by 12, not 6, relative to the log arguments, or rather the coefficients in step 4 are wrong leading to the wrong final constant).
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 4 is mathematically incorrect. The partial fraction decomposition of 1/((2x-1)(2x+2)) is 1/6 * (1/(2x-1) - 1/(2x+2)), which simplifies to 1/(6(2x-1)) - 1/(6(2x+2)). The step claims the coefficients are 1/3 and -1/3, which is off by a factor of 2. Consequently, the final answer is incorrect (it should be divided by 12, not 6, relative to the log arguments, or rather the coefficients in step 4 are wrong leading to the wrong final constant).qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 4 is mathematically incorrect; the partial fraction decomposition of 1/((2x-1)(2x+2)) is 1/6 * (1/(x-1/2) - 1/(x+1)), not 1/(3(2x-1)) - 1/(3(2x+2)). The coefficients in step 4 are wrong (1/3 instead of 1/6 relative to the factored terms, or rather the decomposition was done incorrectly). Specifically, 1/((2x-1)(2x+2)) = A/(2x-1) + B/(2x+2) yields A=1/6, B=-1/6, so the terms should be 1/(6(2x-1)) - 1/(6(2x+2)). Step 4 has 1/3, which is double the correct value. Consequently, the final answer is off by a factor of 2.gpt-oss:20b: fail (error) 2026-10-07 — Step 4 incorrectly decomposes 1/(2*(2*x-1)*(x+1)) as 1/(3*(2*x-1)) - 1/(3*(2*x+2)). The correct partial‑fraction form would involve 1/(x+1), not 1/(2*x+2), and the coefficients are not 1/3. This makes the subsequent steps invalid.
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.