Partial fraction decomposition
Problem 4.394 · hard
Evaluate \( \displaystyle \int \frac{- x^{2} - 10 x + 32}{x^{3} - 2 x^{2} - 4 x + 8}\, dx \) using partial fractions.
- \[ \left(x + 2\right) \left(x - 2\right)^{2} = \left(x - 2\right)^{2} \left(x + 2\right) \]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.
- \[ \frac{- x^{2} - 10 x + 32}{x^{3} - 2 x^{2} - 4 x + 8} = \frac{3}{x + 2} - \frac{4}{x - 2} + \frac{2}{\left(x - 2\right)^{2}} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(- 4 \ln{\left(x - 2 \right)} + 3 \ln{\left(x + 2 \right)} - \frac{2}{x - 2}\right) = \frac{3}{x + 2} - \frac{4}{x - 2} + \frac{2}{\left(x - 2\right)^{2}} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( - 4 \ln{\left(\left|{x - 2}\right| \right)} + 3 \ln{\left(\left|{x + 2}\right| \right)} - \frac{2}{x - 2} + C \)
Lines: 3 proved, 1 not checked. 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 | 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: passqwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution omits absolute value signs in the logarithmic terms (line 4), which is mathematically incorrect for an indefinite integral over a general domain. The stated answer includes them, but the derivation step does not, creating a contradiction and teaching incorrect notation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-05 — [domain objection, downgraded to style] The solution omits absolute value signs in the logarithmic terms (line 4), which is mathematically incorrect for an indefinite integral over a general domain. The stated answer includes them, but the derivation step does not, creating a contradiction and teaching incorrect notation.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly factors the denominator, sets up the partial fraction decomposition for the repeated linear factor, and integrates term-by-term to match the stated answer.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.