Partial fraction decomposition
Problem 4.991 · hard
Evaluate \( \displaystyle \int \frac{- 5 x^{2} + 33 x - 16}{\left(x - 5\right) \left(x - 2\right) \left(x + 3\right)}\, dx \) using partial fractions.
- \[ \left(x + 3\right) \left(x - 2\right) \left(x - 5\right) = \left(x - 5\right) \left(x - 2\right) \left(x + 3\right) \]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.Reviewed
- \[ \frac{- 5 x^{2} + 33 x - 16}{\left(x - 5\right) \left(x - 2\right) \left(x + 3\right)} = - \frac{4}{x + 3} - \frac{2}{x - 2} + \frac{1}{x - 5} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(\ln{\left(x - 5 \right)} - 2 \ln{\left(x - 2 \right)} - 4 \ln{\left(x + 3 \right)}\right) = - \frac{4}{x + 3} - \frac{2}{x - 2} + \frac{1}{x - 5} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( \ln{\left(\left|{x - 5}\right| \right)} - 2 \ln{\left(\left|{x - 2}\right| \right)} - 4 \ln{\left(\left|{x + 3}\right| \right)} + C \)
✓ 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 — The solution correctly identifies the partial fraction decomposition and integrates each term. Although the intermediate equation in step 4 omits absolute value bars in the log arguments, the final stated answer correctly includes them, and the derivative check holds for the domain where the logs are defined.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly identifies the partial fraction decomposition and integrates each term. Although the intermediate equation in step 4 omits absolute value bars in the log arguments, the final stated answer correctly includes them, and the derivative check holds for the domain where the logs are defined.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly identifies the partial fraction decomposition and integrates each term. The use of absolute values in the final answer is appropriate for an indefinite integral, and the intermediate steps are logically sound.gpt-oss:20b: pass 2026-10-12
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 2026-10-12 with SymPy 1.14.0.