Partial fraction decomposition
Problem 4.397 · hard
Evaluate \( \displaystyle \int \frac{4 x^{2} - 12 x + 13}{\left(x - 3\right) \left(x^{2} + 4\right)}\, dx \) using partial fractions.
- \[ \left(x - 3\right) \left(x^{2} + 4\right) \]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
- \[ \frac{4 x^{2} - 12 x + 13}{\left(x - 3\right) \left(x^{2} + 4\right)} = \frac{3 x - 3}{x^{2} + 4} + \frac{1}{x - 3} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(\ln{\left(x - 3 \right)} + \frac{3 \ln{\left(x^{2} + 4 \right)}}{2} - \frac{3 \operatorname{atan}{\left(\frac{x}{2} \right)}}{2}\right) = \frac{3 x - 3}{x^{2} + 4} + \frac{1}{x - 3} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( \frac{3 \ln{\left(x^{2} + 4 \right)}}{2} + \ln{\left(\left|{x - 3}\right| \right)} - \frac{3 \operatorname{atan}{\left(\frac{x}{2} \right)}}{2} + C \)
✓ 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 accurately. The final result matches the stated answer, including the correct handling of the absolute value for the linear factor and the arctangent for the quadratic factor.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the partial fraction decomposition and integrates each term accurately. The final result matches the stated answer, including the correct handling of the absolute value for the linear factor and the arctangent for the quadratic factor.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the partial fraction decomposition and integrates each term accurately, including the absolute value for the linear factor and the arctangent for the quadratic factor.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.