Partial fraction decomposition
Problem 4.531 · hard
Evaluate \( \displaystyle \int \frac{x - 15}{\left(x - 3\right) \left(x + 3\right)}\, dx \) using partial fractions.
- \[ \left(x + 3\right) \left(x - 3\right) = \left(x - 3\right) \left(x + 3\right) \]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.
- \[ \frac{x - 15}{\left(x - 3\right) \left(x + 3\right)} = \frac{3}{x + 3} - \frac{2}{x - 3} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(- 2 \ln{\left(x - 3 \right)} + 3 \ln{\left(x + 3 \right)}\right) = \frac{3}{x + 3} - \frac{2}{x - 3} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( - 2 \ln{\left(\left|{x - 3}\right| \right)} + 3 \ln{\left(\left|{x + 3}\right| \right)} + C \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 (style) — [domain objection, downgraded to style] The solution omits absolute value bars in the logarithmic terms of line 4, which is mathematically incorrect for an indefinite integral over a general domain. Additionally, line 4 uses differentiation to verify the result rather than explicitly performing the integration step, which obscures the method requested.
Every verdict on record (4)
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 bars in the logarithmic terms of line 4, which is mathematically incorrect for an indefinite integral over a general domain. Additionally, line 4 uses differentiation to verify the result rather than explicitly performing the integration step, which obscures the method requested.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the partial fraction decomposition and integrates term-by-term. The final answer matches the stated answer, including the absolute value bars required for the general antiderivative.gpt-oss:20b: pass 2026-10-06
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.