Integral of \( \displaystyle \frac{2 x - 1}{\left(2 x - 1\right)^{2} + 1} \)
Problem 4.1335 · medium Video
Find \( \displaystyle \int \frac{2 x - 1}{\left(2 x - 1\right)^{2} + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{2 x - 1}{\left(2 x - 1\right)^{2} + 1}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{8 x - 4}{4 \left(\left(2 x - 1\right)^{2} + 1\right)}\, dx \]algebraRewrite the numerator to match the derivative of the denominator.✓ Proved
- \[ = \frac{\ln{\left(\left(2 x - 1\right)^{2} + 1 \right)}}{4} \]antiderivativeIntegrate the expression using the logarithmic rule.✓ Proved
Answer \( \frac{\ln{\left(2 x^{2} - 2 x + 1 \right)}}{4} + C \)
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 - 1)**2 + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | lines differ by the constant -log(2)/4 undefined where (2*x - 1)**2 + 1 = 0 log is undefined for non-positive arguments |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: lines differ by the constant log(2)/4 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 2 applies two rules simultaneously: it multiplies by 4/4 (algebra) and factors out the constant 1/4 (linearity). The contract requires each step to change only one thing. Additionally, Step 3 is labeled 'antiderivative' but effectively performs the substitution and integration in one go, skipping the explicit substitution step required by the granularity constraint.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-24 — Step 2 applies two rules simultaneously: it multiplies by 4/4 (algebra) and factors out the constant 1/4 (linearity). The contract requires each step to change only one thing. Additionally, Step 3 is labeled 'antiderivative' but effectively performs the substitution and integration in one go, skipping the explicit substitution step required by the granularity constraint.gpt-oss:20b: pass 2026-10-24qwen3.6:27b-mlx: fail (style) 2026-10-23 — Step 2 applies two operations at once: multiplying by 4/4 (algebra) and factoring out the constant 1/4 (linearity). The contract requires each step to change only one thing; this should be split into a step applying linearity to pull out 1/4, followed by a step rewriting the numerator.gpt-oss:20b: pass 2026-10-23qwen3.6:27b-mlx: fail (error) 2026-10-23 — Step 2 applies two operations simultaneously: multiplying by 4/4 (algebra) and factoring out the constant 1/4 (linearity). The contract requires each step to change only one thing. Additionally, Step 3 is labeled 'antiderivative' but performs the integration; while 'antiderivative' is in the vocabulary, the primary defect is the violation of the single-rule-per-step constraint in Step 2.gpt-oss:20b: pass 2026-10-23
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 2026-10-24 with SymPy 1.14.0.