Properties of definite integrals
Problem 4.497 · easy
Suppose \( \displaystyle \int_0^{3} f(x)\,dx = -9 \), \( \displaystyle \int_0^{2} f(x)\,dx = 1 \), \( \displaystyle \int_0^{3} g(x)\,dx = -1 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 7 \). Find \( \displaystyle \int_{2}^{3} \left(f(x) + 2\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 2 to 3 = ∫ from 0 to 3 − ∫ from 0 to 2.
- \[ 1 \left(-10\right) + 0 \left(-8\right) + 2 \cdot 1 = -8 \]Combine the known values.✓ Proved
Answer \( -8 \)
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 (misleading) — The second line is a garbled equation that does not correctly express the calculation: it mixes unrelated numbers (1, -10, 0, -8, 2) without explaining their origin. A clear derivation would be ∫₂³(f+2)=∫₀³f−∫₀²f+2(3−2)=(-9−1)+2=−8. The current statement could mislead a student about how the result is obtained.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-06 — The second line is a garbled equation that does not correctly express the calculation: it mixes unrelated numbers (1, -10, 0, -8, 2) without explaining their origin. A clear derivation would be ∫₂³(f+2)=∫₀³f−∫₀²f+2(3−2)=(-9−1)+2=−8. The current statement could mislead a student about how the result is obtained.qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer returned a non-objectgpt-oss:20b: fail (misleading) 2026-10-06 — The second sentence mixes unrelated numbers (1,0,2) with the integrals in a confusing way. It does not clearly explain how the value -8 is obtained, potentially misleading a student about the correct application of linearity and the constant term.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly calculates the integral of the constant 2 as 0 instead of 2*(3-2)=2. It also appears to use incorrect values for the integrals of f and g, leading to a wrong intermediate calculation.
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 2026-10-06 with SymPy 1.14.0.