Partial fraction decomposition
Problem 4.693 · hard
Evaluate \( \displaystyle \int \frac{- 3 x^{2} - 8 x - 29}{\left(x - 5\right) \left(x + 1\right)^{2}}\, dx \) using partial fractions.
- \[ \left(x + 1\right)^{2} \left(x - 5\right) = \left(x - 5\right) \left(x + 1\right)^{2} \]Factor the denominator.✓ Proved
- Set up one term per factor (a repeated factor gets a term for each power; an irreducible quadratic gets Bx + C on top), clear denominators, and match coefficients.
- \[ \frac{- 3 x^{2} - 8 x - 29}{\left(x - 5\right) \left(x + 1\right)^{2}} = \frac{1}{x + 1} + \frac{4}{\left(x + 1\right)^{2}} - \frac{4}{x - 5} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(- 4 \ln{\left(x - 5 \right)} + \ln{\left(x + 1 \right)} - \frac{4}{x + 1}\right) = \frac{1}{x + 1} + \frac{4}{\left(x + 1\right)^{2}} - \frac{4}{x - 5} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( - 4 \ln{\left(\left|{x - 5}\right| \right)} + \ln{\left(\left|{x + 1}\right| \right)} - \frac{4}{x + 1} + C \)
Lines: 3 proved, 1 not checked. 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 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | SymPy's own apart() gives the same decomposition, and the answer differentiates back |
Reviewers
gpt-oss:20b: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The solution integrates \(-4\log(x-5)+\log(x+1)-4/(x+1)\) without absolute values in the logarithms, which is incorrect for an indefinite integral ovqwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution omits absolute value bars in the logarithmic terms of the antiderivative (line 4), which is mathematically incorrect for an indefinite integral over the general domain. While the final stated answer includes them, the derived equation in line 4 does not, teaching the student that ln(x-a) is sufficient.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-08 — [domain objection, downgraded to style] The solution omits absolute value bars in the logarithmic terms of the antiderivative (line 4), which is mathematically incorrect for an indefinite integral over the general domain. While the final stated answer includes them, the derived equation in line 4 does not, teaching the student that ln(x-a) is sufficient.gpt-oss:20b: inconclusive 2026-10-08 — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The solution integrates \(-4\log(x-5)+\log(x+1)-4/(x+1)\) without absolute values in the logarithms, which is incorrect for an indefinite integral ovqwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the partial fraction decomposition and integrates each term accurately, including the absolute values in the logarithmic terms as required for indefinite integrals.gpt-oss:20b: pass 2026-10-08
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
generator:structured/partial_fraction_decomp, checked 2026-10-08 with SymPy 1.14.0.