Properties of definite integrals
Problem 4.946 · easy
Suppose \( \displaystyle \int_0^{4} f(x)\,dx = -5 \), \( \displaystyle \int_0^{2} f(x)\,dx = -8 \), \( \displaystyle \int_0^{4} g(x)\,dx = 2 \) and \( \displaystyle \int_0^{2} g(x)\,dx = -3 \). Find \( \displaystyle \int_{4}^{2} \left(3 f(x) + g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 2 to 4 = ∫ from 0 to 4 − ∫ from 0 to 2; swapping the limits changes the sign.
- \[ 3 \left(-3\right) + 1 \left(-5\right) + 0 \left(-2\right) = -14 \]Combine the known values.✓ Proved
Answer \( -14 \)
Lines: 1 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 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ 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 | integrated concrete linear functions built to have the given integrals |
Reviewers
gpt-oss:20b: fail (error) — The second line incorrectly combines the given integrals; it uses the raw values of ∫₀²g and ∫₀⁴f instead of computing the differences needed for ∫₂⁴(3f+g).qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — The second line incorrectly combines the given integrals; it uses the raw values of ∫₀²g and ∫₀⁴f instead of computing the differences needed for ∫₂⁴(3f+g).qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer returned a non-objectgpt-oss:20b: fail (misleading) 2026-10-12 — The second sentence incorrectly combines the given integrals; it uses the wrong values and signs, leading to a false computation. The correct evaluation is 3·3+5=14 for ∫₂⁴(3f+g), hence ∫₄²(3f+g)=−14.qwen3.6:27b-mlx: inconclusive 2026-10-12 — 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/integral_properties, checked 2026-10-12 with SymPy 1.14.0.