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Properties of definite integrals

Problem 4.498 · easy

Suppose \( \displaystyle \int_0^{5} f(x)\,dx = 7 \), \( \displaystyle \int_0^{2} f(x)\,dx = 8 \), \( \displaystyle \int_0^{5} g(x)\,dx = -9 \) and \( \displaystyle \int_0^{2} g(x)\,dx = -4 \). Find \( \displaystyle \int_{0}^{5} \left(g(x)\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out.
\[ 1 \left(-9\right) + 0 \cdot 7 + 0 \cdot 5 = -9 \]
Combine the known values.✓ Proved

Answer \( -9 \)

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: inconclusive — reviewer returned a non-object
qwen3.6:27b-mlx: fail (error) — The solution provides no logical derivation connecting the given premises to the answer. It simply restates the given value for the integral of g(x) using an arbitrary and unexplained linear combination, failing to demonstrate any calculus reasoning.

Every verdict on record (4)

gpt-oss:20b: inconclusive 2026-10-06 — reviewer returned a non-object
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution provides no logical derivation connecting the given premises to the answer. It simply restates the given value for the integral of g(x) using an arbitrary and unexplained linear combination, failing to demonstrate any calculus reasoning.
gpt-oss:20b: fail (misleading) 2026-10-06 — The solution’s second line attempts to ‘combine’ known integrals using 0*7 and 0*5, which is unnecessary and does not follow from the stated property. It misleads the reader into thinking a non‑trivial computation was performed, whereas the result was simply given in the problem statement.
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution provides no logical derivation connecting the given premises to the answer. It simply restates the given value for the integral of g(x) from 0 to 5, ignoring the other data entirely, which makes the reasoning vacuous and fails to demonstrate how to use integral properties.

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.