Integral of \( \displaystyle \frac{1}{\left(2 x - 1\right)^{2} - 4} \)
Problem 4.1267 · medium Video
Find \( \displaystyle \int \frac{1}{\left(2 x - 1\right)^{2} - 4} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(2 x - 1\right)^{2} - 4}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{1}{4 x^{2} - 4 x - 3}\, dx \]algebra simplifyExpand the squared term. Combine the constants.✓ Proved
- \[ = \int \frac{1}{\left(2 x - 3\right) \left(2 x + 1\right)}\, dx \]algebraFactor the quadratic denominator.✓ Proved
- \[ = \int \left(- \frac{1}{4 \left(2 x + 1\right)} + \frac{1}{4 \left(2 x - 3\right)}\right)\, dx \]partial-fractionsPerform partial fraction decomposition.✓ Proved
- \[ = \int \frac{1}{8 x - 12}\, dx - \int \frac{1}{8 x + 4}\, dx \]linearitySplit the integral into two parts.≈ Checked numerically
- \[ = \frac{\int \frac{1}{2 x - 3}\, dx}{4} - \frac{\int \frac{1}{2 x + 1}\, dx}{4} \]linearityFactor out the constant 1/4.✓ Proved
- \[ = \frac{\ln{\left(2 x - 3 \right)}}{8} - \frac{\ln{\left(2 x + 1 \right)}}{8} \]antiderivative simplifyIntegrate each term using the substitution rule for log(ax+b). Simplify the coefficients.✓ Proved
Answer \( \frac{\ln{\left(x - \frac{3}{2} \right)} - \ln{\left(x + \frac{1}{2} \right)}}{8} + C \)
✓ Nihil obstat Lines: 8 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 - 4 = 0 undefined where 4*x**2 - 4*x - 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4*x - 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4*x - 3 = 0 undefined where 2*x - 3 = 0 undefined where 2*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 3 = 0 undefined where 2*x + 1 = 0 |
| 6 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log(x - 3/2)/8 - log(x + 1/2)/8 - log(8*x - 12)/8 + log(8*x + 4)/8; numeric agreement only, at 24 of 24 sampled points undefined where 2*x - 3 = 0 undefined where 2*x + 1 = 0 undefined where 8*x - 12 = 0 undefined where 8*x + 4 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 8*x - 12 = 0 undefined where 8*x + 4 = 0 undefined where 2*x - 3 = 0 undefined where 2*x + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 3 = 0 undefined where 2*x + 1 = 0 log is undefined for non-positive arguments |
| 9 | ✓ 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 - 3/2)/8 + log(x + 1/2)/8 + log(2*x - 3)/8 - log(2*x + 1)/8; 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: pass — The solution correctly applies partial fraction decomposition and integration rules. Each step changes only one aspect of the expression and uses valid labels from the provided vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies partial fraction decomposition and integration rules. Each step changes only one aspect of the expression and uses valid labels from the provided vocabulary.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies algebraic manipulation, partial fraction decomposition, and integration rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies partial fraction decomposition and linearity of integration. The final answer matches the stated answer upon simplification (log(2x-3) = log(2(x-3/2)) = log(2) + log(x-3/2), and the constant log(2) is absorbed into the constant of integration).gpt-oss:20b: pass 2026-10-17
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 2026-10-19 with SymPy 1.14.0.