Partial fraction decomposition
Problem 4.536 · hard
Evaluate \( \displaystyle \int \frac{8 - 4 x}{x^{3} - 11 x^{2} + 39 x - 45}\, dx \) using partial fractions.
- \[ \left(x - 3\right)^{2} \left(x - 5\right) = \left(x - 5\right) \left(x - 3\right)^{2} \]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.Reviewed
- \[ \frac{8 - 4 x}{x^{3} - 11 x^{2} + 39 x - 45} = \frac{3}{x - 3} + \frac{2}{\left(x - 3\right)^{2}} - \frac{3}{x - 5} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(- 3 \ln{\left(x - 5 \right)} + 3 \ln{\left(x - 3 \right)} - \frac{2}{x - 3}\right) = \frac{3}{x - 3} + \frac{2}{\left(x - 3\right)^{2}} - \frac{3}{x - 5} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( - 3 \ln{\left(\left|{x - 5}\right| \right)} + 3 \ln{\left(\left|{x - 3}\right| \right)} - \frac{2}{x - 3} + C \)
Lines: 3 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly factors the denominator, sets up the partial fraction decomposition for the repeated linear factor, and integrates term-by-term. The final result matches the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly factors the denominator, sets up the partial fraction decomposition for the repeated linear factor, and integrates term-by-term. The final result matches the stated answer.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to include absolute value signs in the logarithmic terms of the final answer (line 4), which is required for the general indefinite integral of 1/(x-a). Additionally, the explanation in line 2 mentions irreducible quadratics, which are not present in this problem, making the setup description misleading.
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.