Partial fraction decomposition
Problem 4.844 · hard
Evaluate \( \displaystyle \int \frac{- x^{2} - 30 x + 16}{\left(x - 4\right) \left(x - 1\right) \left(x + 4\right)}\, dx \) using partial fractions.
- \[ \left(x + 4\right) \left(x - 1\right) \left(x - 4\right) = \left(x - 4\right) \left(x - 1\right) \left(x + 4\right) \]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.
- \[ \frac{- x^{2} - 30 x + 16}{\left(x - 4\right) \left(x - 1\right) \left(x + 4\right)} = \frac{3}{x + 4} + \frac{1}{x - 1} - \frac{5}{x - 4} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(- 5 \ln{\left(x - 4 \right)} + \ln{\left(x - 1 \right)} + 3 \ln{\left(x + 4 \right)}\right) = \frac{3}{x + 4} + \frac{1}{x - 1} - \frac{5}{x - 4} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( - 5 \ln{\left(\left|{x - 4}\right| \right)} + \ln{\left(\left|{x - 1}\right| \right)} + 3 \ln{\left(\left|{x + 4}\right| \right)} + C \)
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 (misleading) — Line 4 claims to integrate term by term but presents a differentiation check instead of the integration step. Additionally, the integration of terms like 1/(x-a) requires absolute values inside the logarithm, which are missing in the intermediate expression shown in Line 4, although they appear in the final stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (misleading) 2026-10-10 — Line 4 claims to integrate term by term but presents a differentiation check instead of the integration step. Additionally, the integration of terms like 1/(x-a) requires absolute values inside the logarithm, which are missing in the intermediate expression shown in Line 4, although they appear in the final stated answer.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the partial fraction decomposition and integrates each term, including the necessary absolute value signs in the final answer.
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 2026-10-10 with SymPy 1.14.0.