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Properties of definite integrals

Problem 4.783 · easy

Suppose $ \displaystyle \int_0^{3} f(x)\,dx = -2 $, $ \displaystyle \int_0^{1} f(x)\,dx = 4 $, $ \displaystyle \int_0^{3} g(x)\,dx = 7 $ and $ \displaystyle \int_0^{1} g(x)\,dx = 3 $. Find $ \displaystyle \int_{0}^{1} \left(g(x)\right) dx $.

The integral of a sum is the sum of the integrals, and constants factor out.
\[ 0 \cdot 4 + 0 \cdot 1 + 1 \cdot 3 = 3 \]
Combine the known values.✓ Proved

Answer $ 3 $

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 response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The solution restates the given value of \int_0^1 g(x)dx without justification. The equation 0*4+0*1+1*3=3 is arbitrary and does not follow from any
qwen3.6:27b-mlx: fail (error) — The solution provides no logical derivation connecting the given integrals to the answer. It simply restates the given value for the target integral using an unexplained arithmetic expression, failing to demonstrate any calculus reasoning.

Every verdict on record (4)

gpt-oss:20b: inconclusive 2026-10-09 — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The solution restates the given value of \int_0^1 g(x)dx without justification. The equation 0*4+0*1+1*3=3 is arbitrary and does not follow from any
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution provides no logical derivation connecting the given integrals to the answer. It simply restates the given value for the target integral using an unexplained arithmetic expression, failing to demonstrate any calculus reasoning.
gpt-oss:20b: fail (misleading) 2026-10-09 — The solution’s second line invents a meaningless linear combination of the given integrals (0·4+0·1+1·3) to arrive at the answer. While the final value happens to match the given data, the reasoning is incorrect and would mislead a student into thinking such a combination is justified.
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution provides no logical derivation connecting the given integrals to the answer. It simply restates the value of the requested integral ($\int_0^1 g(x)dx = 3$) as a result of an arbitrary arithmetic calculation ($0*4 + 0*1 + 1*3$) without explaining why this linear combination is relevant or correct.

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 2026-10-09 with SymPy 1.14.0.