Partial fraction decomposition
Problem 4.399 · hard
Evaluate \( \displaystyle \int \frac{x^{2} + 26 x - 80}{x^{3} - 3 x^{2} - 16 x + 48}\, dx \) using partial fractions.
- \[ \left(x + 4\right) \left(x - 3\right) \left(x - 4\right) = \left(x - 4\right) \left(x - 3\right) \left(x + 4\right) \]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.Reviewed
- \[ \frac{x^{2} + 26 x - 80}{x^{3} - 3 x^{2} - 16 x + 48} = - \frac{3}{x + 4} - \frac{1}{x - 3} + \frac{5}{x - 4} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(5 \ln{\left(x - 4 \right)} - \ln{\left(x - 3 \right)} - 3 \ln{\left(x + 4 \right)}\right) = - \frac{3}{x + 4} - \frac{1}{x - 3} + \frac{5}{x - 4} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( 5 \ln{\left(\left|{x - 4}\right| \right)} - \ln{\left(\left|{x - 3}\right| \right)} - 3 \ln{\left(\left|{x + 4}\right| \right)} + C \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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 | 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly factors the denominator, sets up the partial fraction decomposition, and integrates term-by-term. The final answer matches the stated result.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.