Integral of \( \displaystyle 2 x - 1 + \frac{1}{2 x - 1} \)
Problem 4.927 · medium
Find \( \displaystyle \int 2 x - 1 + \frac{1}{2 x - 1} \, dx \). (Omit the constant of integration.)
- \[ \int \left(2 x - 1 + \frac{1}{2 x - 1}\right)\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = - \int 1\, dx + \int 2 x\, dx + \int \frac{1}{2 x - 1}\, dx \]linearitySplit the integral into three separate parts.✓ Proved
- \[ = x^{2} - x + \int \frac{1}{2 x - 1}\, dx \]antiderivative rewriteEvaluate the first two polynomial terms. Factor out the 2 from the denominator.✓ Proved
- \[ = x^{2} - x + \frac{\int \frac{1}{x - \frac{1}{2}}\, dx}{2} \]algebraMove the constant 1/2 outside the integral.✓ Proved
- \[ = x^{2} - x + \frac{\ln{\left(2 x - 1 \right)}}{2} \]antiderivativeIntegrate the remaining term using the logarithmic rule.✓ Proved
Answer \( x^{2} - x + \frac{\ln{\left(2 x - 1 \right)}}{2} + C \)
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.
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 x - 1/2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1/2 = 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 (error) — Step 6 is mathematically incorrect: the antiderivative of 1/(x - 1/2) is log(x - 1/2), not log(2*x - 1). The step skips the necessary substitution or constant adjustment, resulting in an incorrect final term (it should be 1/2 * log(x - 1/2) or log(2*x - 1)/2, but the step claims the integral of 1/(x-1/2) is log(2*x-1), which is false).
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 6 is mathematically incorrect: the antiderivative of 1/(x - 1/2) is log(x - 1/2), not log(2*x - 1). The step skips the necessary substitution or constant adjustment, resulting in an incorrect final term (it should be 1/2 * log(x - 1/2) or log(2*x - 1)/2, but the step claims the integral of 1/(x-1/2) is log(2*x-1), which is false).qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 6 is mathematically incorrect. The integral of 1/(x - 1/2) is log(x - 1/2), not log(2*x - 1). The step skips the necessary substitution or adjustment for the constant factor inside the logarithm, resulting in an incorrect final term (it should be 1/2 * log(x - 1/2) or equivalently 1/2 * log(2*x - 1) - 1/2 * log(2), but the step claims the integral of 1/(x-1/2) directly yields log(2x-1) which is false).gpt-oss:20b: pass 2026-10-12
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 2026-10-12 with SymPy 1.14.0.