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Properties of definite integrals

Problem 4.495 · easy

Suppose \( \displaystyle \int_0^{6} f(x)\,dx = 7 \), \( \displaystyle \int_0^{2} f(x)\,dx = -6 \), \( \displaystyle \int_0^{6} g(x)\,dx = 5 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 4 \). Find \( \displaystyle \int_{0}^{6} \left(g(x)\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out.
\[ 0 \cdot 7 + 0 \cdot 6 + 1 \cdot 5 = 5 \]
Combine the known values.✓ Proved

Answer \( 5 \)

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in 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: pass
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer returned a non-object
gpt-oss:20b: fail (error) 2026-10-06 — The solution’s second line arbitrarily sets coefficients 0,0,1 and claims to combine known values, but provides no justification for why these coefficients are chosen or how they relate to the desired integral. This step does not follow from the preceding statement and misleads the reader into thinking a non‑existent linear combination argument is being used.
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution is completely nonsensical. It ignores the provided information about f(x) and the split of the interval, and instead performs an arbitrary arithmetic operation (0*7 + 0*6 + 1*5) that has no logical connection to the problem statement, even though it coincidentally yields the correct value for the integral of g(x) which was already given in the problem.

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.