Integrals of absolute values
Problem 4.745 · easy
Evaluate \( \displaystyle \int_{-5}^{0} \left| - x^{2} - 4 x - 3 \right| dx \).
- \[ - x^{2} - 4 x - 3 = \left(- x - 1\right) \left(x + 3\right) \]The integrand is zero at x = -3, x = -1.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).
- \[ \int\limits_{-3}^{-1} \left(- x^{2} - 4 x - 3\right)\, dx + \int\limits_{-1}^{0} \left(x^{2} + 4 x + 3\right)\, dx + \int\limits_{-5}^{-3} \left(x^{2} + 4 x + 3\right)\, dx = \frac{28}{3} \]Integrate each piece and add.✓ Proved
Answer \( \frac{28}{3} \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | numerical quadrature of |p(x)| |
Reviewers
gpt-oss:20b: fail (misleading) — The sentence claims the sign does not change between the zeros, which is incorrect; the polynomial changes sign at each root. The rest of the solution is correct.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-09 — The sentence claims the sign does not change between the zeros, which is incorrect; the polynomial changes sign at each root. The rest of the solution is correct.qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-09 — The solution incorrectly states that the sign does not change between the zeros, and it integrates the original polynomial on the interval (-3,-1) where the integrand should be its positive value. The sign analysis and the choice of integrand on each subinterval are wrong, leading to an incorrect result.qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-object
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/absolute_value_integral, checked 2026-10-09 with SymPy 1.14.0.