Partial fraction decomposition
Problem 4.532 · hard
Evaluate \( \displaystyle \int \frac{3 x - 7}{x^{2} - 6 x + 5}\, dx \) using partial fractions.
- \[ \left(x - 1\right) \left(x - 5\right) = \left(x - 5\right) \left(x - 1\right) \]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.
- \[ \frac{3 x - 7}{x^{2} - 6 x + 5} = \frac{1}{x - 1} + \frac{2}{x - 5} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(2 \ln{\left(x - 5 \right)} + \ln{\left(x - 1 \right)}\right) = \frac{1}{x - 1} + \frac{2}{x - 5} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( 2 \ln{\left(\left|{x - 5}\right| \right)} + \ln{\left(\left|{x - 1}\right| \right)} + C \)
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 integration step (line 4), which is mathematically incorrect for the general indefinite integral. While the final stated answer includes them, the derivation shown is invalid for x < 1 or 1 < x < 5.
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 integration step (line 4), which is mathematically incorrect for the general indefinite integral. While the final stated answer includes them, the derivation shown is invalid for x < 1 or 1 < x < 5.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (style) 2026-10-06 — [domain objection, downgraded to style] The solution omits absolute value signs in the integration step (line 4), which is mathematically incorrect for indefinite integrals of 1/(x-a) unless the domain is restricted. While the final stated answer includes them, the derivation shown is flawed and would teach a student that absolute values are optional.
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.