Integral of \( \displaystyle \frac{1}{\left(2 x + 1\right)^{2} - 4} \)
Problem 4.805 · medium
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
- \[ = \int \frac{1}{4 x^{2} + 4 x - 3}\, dx \]algebraExpand the denominator.✓ Proved
- \[ = \int \frac{1}{\left(2 x - 1\right) \left(2 x + 3\right)}\, dx \]algebraFactor the quadratic expression.✓ Proved
- \[ = \int \left(- \frac{1}{4 \left(2 x + 3\right)} + \frac{1}{4 \left(2 x - 1\right)}\right)\, dx \]partial-fractionsPerform partial fraction decomposition.✓ Proved
- \[ = \frac{\int \frac{1}{2 x - 1}\, dx}{4} - \frac{\int \frac{1}{2 x + 3}\, dx}{4} \]linearitySplit the integral into two parts.≈ Checked numerically
- \[ = \frac{\ln{\left(2 x - 1 \right)}}{8} - \frac{\ln{\left(2 x + 3 \right)}}{8} \]antiderivative algebraIntegrate each term using the substitution rule for log(ax+b). Simplify the coefficients.✓ Proved
Answer \( \frac{\ln{\left(x - \frac{1}{2} \right)} - \ln{\left(x + \frac{3}{2} \right)}}{8} + C \)
✓ Nihil obstat Lines: 6 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 undefined where 2*x - 1 = 0 undefined where 2*x + 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 undefined where 2*x + 3 = 0 |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log(x - 1/2)/8 - log(x + 3/2)/8 - log(2*x - 1)/8 + log(2*x + 3)/8; numeric agreement only, at 24 of 24 sampled points undefined where 2*x + 3 = 0 undefined where 2*x - 1 = 0 |
| 6 | ✓ 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 |
| 7 | ✓ 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 - 1/2)/8 + log(x + 3/2)/8 + log(2*x - 1)/8 - log(2*x + 3)/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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies algebraic expansion, partial fraction decomposition, linearity of integration, and standard antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-10
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 2026-10-10 with SymPy 1.14.0.