Integral of \( \displaystyle \frac{1}{x \left(x + 1\right)^{2}} \)
Problem 4.185 · medium
Find \( \displaystyle \int \frac{1}{x \left(x + 1\right)^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{x \left(x + 1\right)^{2}}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(- \frac{1}{x + 1} - \frac{1}{\left(x + 1\right)^{2}} + \frac{1}{x}\right)\, dx \]partial-fractionsPerform partial fraction decomposition.✓ Proved
- \[ = \int \frac{1}{x}\, dx - \int \frac{1}{\left(x + 1\right)^{2}}\, dx - \int \frac{1}{x + 1}\, dx \]linearitySplit the integral into three separate terms.✓ Proved
- \[ = \ln{\left(x \right)} - \ln{\left(x + 1 \right)} + \frac{1}{x + 1} \]antiderivativeIntegrate each term separately.✓ Proved
- \[ = \ln{\left(\frac{x}{x + 1} \right)} + \frac{1}{x + 1} \]simplifyCombine the logarithmic terms using log(a) - log(b) = log(a/b).≈ Checked numerically
Answer \( \ln{\left(x \right)} - \ln{\left(x + 1 \right)} + \frac{1}{x + 1} + C \)
Lines: 4 proved, 2 checked numerically. 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 + 1 = 0 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x = 0 log is undefined for non-positive arguments |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log(x) - log(x/(x + 1)) - log(x + 1); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where x + 1 = 0 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -log(x) + log(x/(x + 1)) + log(x + 1); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where x + 1 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: fail (error) — Step 4 applies the antiderivative rule incorrectly: ∫1/(x+1)²dx = –1/(x+1), not +1/(x+1). The sign of the last term is wrong.qwen3.6:27b-mlx: pass — The solution correctly applies partial fraction decomposition, linearity of integration, and standard antiderivatives. The final simplification is algebraically valid and clearly labeled.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies partial fraction decomposition, linearity of integration, and standard antiderivatives. The final simplification is algebraically valid and clearly labeled.gpt-oss:20b: fail (error) 2026-09-29 — Step 4 applies the antiderivative rule incorrectly: ∫1/(x+1)²dx = –1/(x+1), not +1/(x+1). The sign of the last term is wrong.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies partial fraction decomposition, linearity of integration, and standard antiderivatives. The final simplification is algebraically valid and clearly labeled.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.