Integral of \( \displaystyle \frac{4 x - 1}{2 x + \left(2 x - 1\right)^{2} + 2} \)
Problem 4.1127 · hard Video
Find \( \displaystyle \int \frac{4 x - 1}{2 x + \left(2 x - 1\right)^{2} + 2} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{4 x - 1}{2 x + \left(2 x - 1\right)^{2} + 2}\, dx \]integralStart with the given integral.✓ Proved
- \[ = \int \frac{4 x - 1}{4 x^{2} - 2 x + 3}\, dx \]algebra simplify rewriteExpand the squared term. Combine like terms in the denominator. Rewrite the numerator to match the derivative of the denominator.✓ Proved
- \[ = - \int 0\, dx + 2 \int \frac{2 x - \frac{1}{2}}{4 x^{2} - 2 x + 3}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = \int \frac{4 x - 1}{4 x^{2} - 2 x + 3}\, dx \]simplify rewriteThe second part is zero. Express the numerator as half of the derivative of the denominator.✓ Proved
- \[ = \frac{\ln{\left(4 x^{2} - 2 x + 3 \right)}}{2} \]antiderivativeIntegrate using the substitution u = 4x^2 - 2x + 3.✓ Proved
Answer \( \frac{\ln{\left(4 x^{2} - 2 x + 3 \right)}}{2} + C \)
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 2*x + (2*x - 1)**2 + 2 = 0 undefined where 4*x**2 - 2*x + 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 2*x + 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 2*x + 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 2*x + 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 2*x + 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 2*x + 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 2*x + 3 = 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: fail (style) — Step 5 applies linearity to split the integral but simultaneously drops the factor of 2 from the first term (changing 2*(...) to just (...)), applying two rules at once. Step 6 is logically circular, claiming the result of Step 5 simplifies back to the integrand of Step 3.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-17 — Step 5 applies linearity to split the integral but simultaneously drops the factor of 2 from the first term (changing 2*(...) to just (...)), applying two rules at once. Step 6 is logically circular, claiming the result of Step 5 simplifies back to the integrand of Step 3.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 5 incorrectly applies the linearity rule by splitting the numerator into two separate integrals without properly distributing the constant factor, resulting in a mathematically invalid expression (the term '2 * Integral(...)' is not equivalent to the previous step's integrand structure). Furthermore, Step 6 reverts to the original integrand, making the intermediate steps 4 and 5 logically incoherent and redundant.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 4 is mathematically incorrect: it claims 4*x - 1 equals 2*(2*x - 1/2) + 0, but 2*(2*x - 1/2) simplifies to 4*x - 1, so the step is actually angpt-oss:20b: fail (style) 2026-10-14 — Step 5 applies two rules at once: it splits the integrand into two parts and then factors a constant out of one of them. Each step should change only one thing, so this step violates the granularity rule.
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 2026-10-17 with SymPy 1.14.0.