Partial fraction decomposition
Problem 4.692 · hard
Evaluate \( \displaystyle \int \frac{x^{2} - 17 x + 50}{\left(x - 5\right)^{2} \left(x - 3\right)}\, dx \) using partial fractions.
- \[ \left(x - 3\right) \left(x - 5\right)^{2} = \left(x - 5\right)^{2} \left(x - 3\right) \]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.
- \[ \frac{x^{2} - 17 x + 50}{\left(x - 5\right)^{2} \left(x - 3\right)} = \frac{2}{x - 3} - \frac{1}{x - 5} - \frac{5}{\left(x - 5\right)^{2}} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(- \ln{\left(x - 5 \right)} + 2 \ln{\left(x - 3 \right)} + \frac{5}{x - 5}\right) = \frac{2}{x - 3} - \frac{1}{x - 5} - \frac{5}{\left(x - 5\right)^{2}} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( - \ln{\left(\left|{x - 5}\right| \right)} + 2 \ln{\left(\left|{x - 3}\right| \right)} + \frac{5}{x - 5} + C \)
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 include the constant of integration (+ C) in the final result, which is required for an indefinite integral. Additionally, the absolute value bars are missing from the logarithmic terms in the stated answer, which is mathematically incorrect for general real x.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to include the constant of integration (+ C) in the final result, which is required for an indefinite integral. Additionally, the absolute value bars are missing from the logarithmic terms in the stated answer, which is mathematically incorrect for general real x.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution skips the actual derivation of the partial fraction coefficients (Step 3 is asserted without showing the work to find A, B, and C). Furthermore, Step 4 claims to integrate term-by-term but presents a derivative check instead of the integration steps, and omits the absolute value bars in the logarithmic terms within the derivative expression, which is technically imprecise for indefinite integrals.gpt-oss:20b: pass 2026-10-08
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 2026-10-08 with SymPy 1.14.0.