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Partial fraction decomposition

Problem 4.691 · hard

Evaluate \( \displaystyle \int \frac{16}{x^{2} - 6 x + 5}\, dx \) 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.
\[ \frac{16}{x^{2} - 6 x + 5} = - \frac{4}{x - 1} + \frac{4}{x - 5} \]
The decomposition; recombining it gives back the original fraction.✓ Proved
\[ \frac{d}{d x} \left(4 \ln{\left(x - 5 \right)} - 4 \ln{\left(x - 1 \right)}\right) = - \frac{4}{x - 1} + \frac{4}{x - 5} \]
Integrate term by term (log|x − a| for each linear factor).✓ Proved

Answer \( 4 \ln{\left(\left|{x - 5}\right| \right)} - 4 \ln{\left(\left|{x - 1}\right| \right)} + C \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution omits absolute value bars in the logarithmic terms (line 4), which is required for the general indefinite integral of 1/(x-a). While the stated answer includes them, the derivation shown is technically incorrect for domains where x-5 or x-1 are negative.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution omits absolute value bars in the logarithmic terms (line 4), which is required for the general indefinite integral of 1/(x-a). While the stated answer includes them, the derivation shown is technically incorrect for domains where x-5 or x-1 are negative.
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: fail (style) 2026-10-08 — [domain objection, downgraded to style] The solution omits absolute value signs in the logarithmic terms (line 4), which is mathematically incorrect for an indefinite integral over a general domain. While the final stated answer includes them, the derivation shown is flawed and would teach a student to ignore the domain restrictions of the logarithm.
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.