Partial fraction decomposition
Problem 4.988 · hard
Evaluate \( \displaystyle \int \frac{35 - 3 x^{2}}{x^{3} + 5 x^{2} + 3 x - 9}\, dx \) using partial fractions.
- \[ \left(x + 3\right)^{2} \left(x - 1\right) = \left(x - 1\right) \left(x + 3\right)^{2} \]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.
- \[ \frac{35 - 3 x^{2}}{x^{3} + 5 x^{2} + 3 x - 9} = - \frac{5}{x + 3} - \frac{2}{\left(x + 3\right)^{2}} + \frac{2}{x - 1} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(2 \ln{\left(x - 1 \right)} - 5 \ln{\left(x + 3 \right)} + \frac{2}{x + 3}\right) = - \frac{5}{x + 3} - \frac{2}{\left(x + 3\right)^{2}} + \frac{2}{x - 1} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( 2 \ln{\left(\left|{x - 1}\right| \right)} - 5 \ln{\left(\left|{x + 3}\right| \right)} + \frac{2}{x + 3} + C \)
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: passqwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to include the constant of integration (+ C) in the final result, which is required for an indefinite integral. Additionally, the logarithmic terms should technically include absolute values to be fully general, though the derivative check holds where defined.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-12 — [domain objection, downgraded to style] The solution fails to include the constant of integration (+ C) in the final result, which is required for an indefinite integral. Additionally, the logarithmic terms should technically include absolute values to be fully general, though the derivative check holds where defined.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The integration of the term -2/(x + 3)^2 is incorrect. The integral of -2(x+3)^(-2) is 2(x+3)^(-1), but the solution presents -5 log|x+3| + 2/(x+3). The -5 log|x+3| term correctly comes from integrating -5/(x+3), but the solution fails to explicitly separate the integration of the distinct partial fraction terms, leading to a confusing and technically incorrect derivation step where the log term is associated with the wrong source or the steps are conflated. Specifically, step 4 claims the derivative of the result equals the sum, which is true, but the sentence 'Integrate term by term' implies a direct mapping that is obscured by the combined result. However, the primary error is that the solution does not show the integration of the squared term correctly in the text description, although the final answer is correct. Wait, let's look closer. The final answer is correct. The derivative check in line 4 proves the answer is correct. The issue is the sentence in line 4: 'Integrate term by term (log|x − a| for each linear factor)'. This sentence is misleading because it ignores the power rule needed for the squared term, implying all terms integrate to logs. This teaches a false method.gpt-oss:20b: pass 2026-10-12
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 2026-10-12 with SymPy 1.14.0.