Partial fraction decomposition
Problem 4.838 · hard
Evaluate \( \displaystyle \int \frac{- 6 x^{2} + 19 x + 1}{x^{3} - 4 x^{2} + 3 x - 12}\, dx \) using partial fractions.
- \[ \left(x - 4\right) \left(x^{2} + 3\right) \]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.
- \[ \frac{- 6 x^{2} + 19 x + 1}{x^{3} - 4 x^{2} + 3 x - 12} = \frac{- 5 x - 1}{x^{2} + 3} - \frac{1}{x - 4} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(- \ln{\left(x - 4 \right)} - \frac{5 \ln{\left(x^{2} + 3 \right)}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}}{3}\right) = \frac{- 5 x - 1}{x^{2} + 3} - \frac{1}{x - 4} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( - \frac{5 \ln{\left(x^{2} + 3 \right)}}{2} - \ln{\left(\left|{x - 4}\right| \right)} - \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} + C \)
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 (error) — The solution fails to explicitly integrate the term (-5x - 1)/(x^2 + 3), which requires splitting into a logarithmic part and an arctangent part. It jumps directly to the final answer without showing the integration steps for the quadratic factor, making the derivation incomplete and potentially confusing regarding the origin of the arctangent term.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to explicitly integrate the term (-5x - 1)/(x^2 + 3), which requires splitting into a logarithmic part and an arctangent part. It jumps directly to the final answer without showing the integration steps for the quadratic factor, making the derivation incomplete and potentially confusing regarding the origin of the arctangent term.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to explicitly handle the integration of the irreducible quadratic term (-5x - 1)/(x^2 + 3). It jumps from the decomposition to the final answer without showing the necessary split into a logarithmic part (from -5x/(x^2+3)) and an arctangent part (from -1/(x^2+3)), nor does it justify the coefficients in the final arctangent term.
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.