Integrals of absolute values
Problem 4.436 · easy
Evaluate \( \displaystyle \int_{-4}^{5} \left| - x^{2} + 2 x + 3 \right| dx \).
- \[ - x^{2} + 2 x + 3 = \left(3 - x\right) \left(x + 1\right) \]The integrand is zero at x = -1, x = 3.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).Reviewed
- \[ \int\limits_{3}^{5} \left(x^{2} - 2 x - 3\right)\, dx + \int\limits_{-1}^{3} \left(- x^{2} + 2 x + 3\right)\, dx + \int\limits_{-4}^{-1} \left(x^{2} - 2 x - 3\right)\, dx = \frac{145}{3} \]Integrate each piece and add.✓ Proved
Answer \( \frac{145}{3} \)
✓ Nihil obstat Lines: 2 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | numerical quadrature of |p(x)| |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the roots of the quadratic, determines the sign of the integrand on the sub-intervals, and sets up the piecewise integration correctly. The final arithmetic is verified by the CAS.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the roots of the quadratic, determines the sign of the integrand on the sub-intervals, and sets up the piecewise integration correctly. The final arithmetic is verified by the CAS.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the roots of the quadratic, determines the sign of the integrand in each interval, and sets up the piecewise integration correctly. The final result is accurate.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/absolute_value_integral, checked 2026-10-05 with SymPy 1.14.0.