Partial fraction decomposition
Problem 4.538 · hard
Evaluate \( \displaystyle \int \frac{- 4 x^{2} + 27 x - 40}{x^{3} - 8 x^{2} + 21 x - 18}\, dx \) using partial fractions.
- \[ \left(x - 2\right) \left(x - 3\right)^{2} = \left(x - 3\right)^{2} \left(x - 2\right) \]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.
- \[ \frac{- 4 x^{2} + 27 x - 40}{x^{3} - 8 x^{2} + 21 x - 18} = - \frac{2}{x - 2} - \frac{2}{x - 3} + \frac{5}{\left(x - 3\right)^{2}} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(- 2 \ln{\left(x - 3 \right)} - 2 \ln{\left(x - 2 \right)} - \frac{5}{x - 3}\right) = - \frac{2}{x - 2} - \frac{2}{x - 3} + \frac{5}{\left(x - 3\right)^{2}} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( - 2 \ln{\left(\left|{x - 3}\right| \right)} - 2 \ln{\left(\left|{x - 2}\right| \right)} - \frac{5}{x - 3} + C \)
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 (misleading) — The solution omits absolute value signs in the logarithmic terms of the final answer, which is required for the general indefinite integral of 1/(x-a). Additionally, the explanation in step 4 incorrectly states that the derivative of the result equals the decomposed fraction, rather than stating that the result is the antiderivative.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (misleading) 2026-10-06 — The solution omits absolute value signs in the logarithmic terms of the final answer, which is required for the general indefinite integral of 1/(x-a). Additionally, the explanation in step 4 incorrectly states that the derivative of the result equals the decomposed fraction, rather than stating that the result is the antiderivative.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The final answer is missing the absolute value bars inside the logarithms, which are required for the general antiderivative of 1/x. Additionally, the solution skips the actual partial fraction decomposition steps, jumping straight to the result.
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 2026-10-06 with SymPy 1.14.0.