Integral of \( \displaystyle \frac{1}{\left(2 x - 1\right)^{2}} \)
Problem 4.1287 · medium Video
Find \( \displaystyle \int \frac{1}{\left(2 x - 1\right)^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(2 x - 1\right)^{2}}\, dx \]integral algebra linearityStart with the integral of the function. Rewrite the negative exponent as a fraction. The constant factor is 1, so no change is needed.✓ Proved
- \[ = - \frac{1}{2 \left(2 x - 1\right)} \]substitutionApply the power rule for integration with the chain rule factor 1/2.✓ Proved
- \[ = - \frac{1}{4 x - 2} \]simplifySimplify the expression.✓ Proved
Answer \( - \frac{1}{4 x - 2} + C \)
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 - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 undefined where 4*x - 2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 4*x - 2 = 0 |
| 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 a logical error: it reverts the expression to the form in Step 1 and labels it 'linearity', which is not a valid transformation rule for this context. Step 4 applies the power rule and chain rule adjustment simultaneously, violating the 'one rule per step' constraint.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-20 — Step 3 is a logical error: it reverts the expression to the form in Step 1 and labels it 'linearity', which is not a valid transformation rule for this context. Step 4 applies the power rule and chain rule adjustment simultaneously, violating the 'one rule per step' constraint.gpt-oss:20b: pass 2026-10-20qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 3 is a logical error: it reverts the expression to the form in Step 1 and labels it 'linearity', which is incorrect because no linear operation was performed to justify the change (or lack thereof). Furthermore, Step 4 applies the power rule for integration, but the label 'substitution' is misleading; while substitution is the underlying mechanism, the step is better described as applying the antiderivative rule directly, and the previous step's nonsense makes the flow invalid.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 3 is a logical error: it reverts the expression to the form in Step 1 but labels it 'linearity', which does not justify reversing the algebraic rewrite from Step 2. Furthermore, Step 4 applies the power rule and the chain rule adjustment simultaneously, violating the single-rule-per-step constraint.gpt-oss:20b: pass 2026-10-19
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 2026-10-20 with SymPy 1.14.0.