Integral of \( \displaystyle \frac{1}{\left(x + 1\right) \left(x + 2\right)^{2}} \)
Problem 4.643 · medium
Find \( \displaystyle \int \frac{1}{\left(x + 1\right) \left(x + 2\right)^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(x + 1\right) \left(x + 2\right)^{2}}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(- \frac{1}{x + 2} - \frac{1}{\left(x + 2\right)^{2}} + \frac{1}{x + 1}\right)\, dx \]partial-fractionsApply partial fraction decomposition.✓ Proved
- \[ = \int \frac{1}{x + 1}\, dx - \int \frac{1}{\left(x + 2\right)^{2}}\, dx - \int \frac{1}{x + 2}\, dx \]linearitySplit the integral into three separate parts.✓ Proved
- \[ = \ln{\left(x + 1 \right)} - \ln{\left(x + 2 \right)} - \int \frac{1}{\left(x + 2\right)^{2}}\, dx \]antiderivativeIntegrate the first two terms.✓ Proved
- \[ = \ln{\left(x + 1 \right)} - \ln{\left(x + 2 \right)} + \frac{1}{x + 2} \]antiderivativeIntegrate the power term.✓ Proved
Answer \( \ln{\left(x + 1 \right)} - \ln{\left(x + 2 \right)} + \frac{1}{x + 2} + C \)
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.
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 + 2 = 0 undefined where x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 2 = 0 undefined where x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 2 = 0 undefined where x + 1 = 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| 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 undefined where 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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The integral of \(1/(x+2)^2\) is \(-1/(x+2)\), not \(+1/(x+2)\). The sign in step 5 is incorrect, leading to an incorrect antiderivative."}qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: inconclusive 2026-10-08 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The integral of \(1/(x+2)^2\) is \(-1/(x+2)\), not \(+1/(x+2)\). The sign in step 5 is incorrect, leading to an incorrect antiderivative."}qwen3.6:27b-mlx: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies 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: inconclusive 2026-10-08 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The integral of \(1/(x+2)^2\) is \(-1/(x+2)\), not \(+1/(x+2)\). The final sign is incorrect, leading to an incorrect antiderivative."}
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 2026-10-08 with SymPy 1.14.0.