Partial fraction decomposition
Problem 4.391 · hard
Evaluate \( \displaystyle \int \frac{x - 4}{\left(x - 3\right) \left(x - 2\right)}\, dx \) using partial fractions.
- \[ \left(x - 2\right) \left(x - 3\right) = \left(x - 3\right) \left(x - 2\right) \]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.
- \[ \frac{x - 4}{\left(x - 3\right) \left(x - 2\right)} = \frac{2}{x - 2} - \frac{1}{x - 3} \]The decomposition; recombining it gives back the original fraction.✓ Proved
- \[ \frac{d}{d x} \left(- \ln{\left(x - 3 \right)} + 2 \ln{\left(x - 2 \right)}\right) = \frac{2}{x - 2} - \frac{1}{x - 3} \]Integrate term by term (log|x − a| for each linear factor).✓ Proved
Answer \( - \ln{\left(\left|{x - 3}\right| \right)} + 2 \ln{\left(\left|{x - 2}\right| \right)} + C \)
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: passqwen3.6:27b-mlx: fail (misleading) — Step 4 claims to integrate term by term but presents the derivative of the result as the equation, which is a verification step, not the integration step itself. Furthermore, it omits the absolute value signs in the logarithmic arguments, which are required for the general indefinite integral.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (misleading) 2026-10-05 — Step 4 claims to integrate term by term but presents the derivative of the result as the equation, which is a verification step, not the integration step itself. Furthermore, it omits the absolute value signs in the logarithmic arguments, which are required for the general indefinite integral.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (misleading) 2026-10-05 — The solution omits absolute value signs in the final antiderivative (Step 4), which is mathematically incorrect for the general indefinite integral. Additionally, Step 1 is redundant since the denominator is already factored.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.