Integral of \( \displaystyle \frac{1}{\left(x + 1\right)^{2} - 4} \)
Problem 4.881 · medium
Find \( \displaystyle \int \frac{1}{\left(x + 1\right)^{2} - 4} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(x + 1\right)^{2} - 4}\, dx \]integralStart with the given integral.✓ Proved
- \[ = \int \frac{1}{x^{2} + 2 x - 3}\, dx \]algebraExpand the squared term.✓ Proved
- \[ = \int \frac{1}{\left(x - 1\right) \left(x + 3\right)}\, dx \]algebraFactor the quadratic expression.✓ Proved
- \[ = \int \left(- \frac{1}{4 \left(x + 3\right)} + \frac{1}{4 \left(x - 1\right)}\right)\, dx \]partial-fractionsPerform partial fraction decomposition.✓ Proved
- \[ = \frac{\ln{\left(x - 1 \right)}}{4} - \frac{\ln{\left(x + 3 \right)}}{4} \]antiderivativeIntegrate each term separately.✓ Proved
Answer \( \frac{\ln{\left(x - 1 \right)} - \ln{\left(x + 3 \right)}}{4} + C \)
✓ Nihil obstat 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. 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 (x + 1)**2 - 4 = 0 undefined where x**2 + 2*x - 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 2*x - 3 = 0 undefined where x - 1 = 0 undefined where x + 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 0 undefined where x + 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 0 undefined where x + 3 = 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: pass — The solution correctly applies algebraic expansion, factoring, partial fraction decomposition, and term-by-term integration. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies algebraic expansion, factoring, partial fraction decomposition, and term-by-term integration. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies algebraic expansion, factoring, partial fraction decomposition, and term-by-term integration. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.