Partial fraction decomposition
Problem 4.392 · hard
Evaluate \( \displaystyle \int \frac{- 9 x^{2} + 32 x - 20}{\left(x - 4\right) \left(x - 1\right)^{2}}\, dx \) using partial fractions.
- \[ \left(x - 1\right)^{2} \left(x - 4\right) = \left(x - 4\right) \left(x - 1\right)^{2} \]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.
- \[ \frac{- 9 x^{2} + 32 x - 20}{\left(x - 4\right) \left(x - 1\right)^{2}} = - \frac{5}{x - 1} - \frac{1}{\left(x - 1\right)^{2}} - \frac{4}{x - 4} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(- 4 \ln{\left(x - 4 \right)} - 5 \ln{\left(x - 1 \right)} + \frac{1}{x - 1}\right) = - \frac{5}{x - 1} - \frac{1}{\left(x - 1\right)^{2}} - \frac{4}{x - 4} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( - 4 \ln{\left(\left|{x - 4}\right| \right)} - 5 \ln{\left(\left|{x - 1}\right| \right)} + \frac{1}{x - 1} + C \)
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: fail (style) — [domain objection, downgraded to style] The solution omits the absolute value bars in the logarithms, which is essential for the antiderivative to be valid on all intervals where the integrand is defined. This could mislead a student into thinking the result is correct without the |·|, potentially causing errors when evaluating the integral across the singularities at x=1 and x=4.qwen3.6:27b-mlx: pass — The solution correctly identifies the partial fraction decomposition form and verifies the coefficients. The integration steps are correct, and the final result matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the partial fraction decomposition form and verifies the coefficients. The integration steps are correct, and the final result matches the stated answer.gpt-oss:20b: fail (style) 2026-10-05 — [domain objection, downgraded to style] The solution omits the absolute value bars in the logarithms, which is essential for the antiderivative to be valid on all intervals where the integrand is defined. This could mislead a student into thinking the result is correct without the |·|, potentially causing errors when evaluating the integral across the singularities at x=1 and x=4.qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the partial fraction decomposition form and the resulting coefficients. The integration steps are correct, and the final answer matches the stated answer.gpt-oss:20b: inconclusive 2026-10-05 — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The solution omits the absolute value bars in the logarithm terms, which could mislead a student into thinking \(\ln(x-4)\) is acceptable without the
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.