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Properties of definite integrals

Problem 4.627 · easy

Suppose \( \displaystyle \int_0^{4} f(x)\,dx = 8 \), \( \displaystyle \int_0^{1} f(x)\,dx = -6 \), \( \displaystyle \int_0^{4} g(x)\,dx = 1 \) and \( \displaystyle \int_0^{1} g(x)\,dx = -4 \). Find \( \displaystyle \int_{1}^{0} \left(2 f(x) - 3 g(x)\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out; swapping the limits changes the sign.
\[ \left(-3\right) 4 + 0 \left(-1\right) + 2 \cdot 6 = 0 \]
Combine the known values.✓ Proved

Answer \( 0 \)

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 substitutes the given integral values. It uses -3*4 instead of -3*(-4) and 2*6 instead of 2*8, and the term 0*(-1) is meaningless. The calculation does not correctly apply the linearity of the integral, leading to an incorrect result.
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (4)

gpt-oss:20b: fail (error) 2026-10-07 — The second line incorrectly substitutes the given integral values. It uses -3*4 instead of -3*(-4) and 2*6 instead of 2*8, and the term 0*(-1) is meaningless. The calculation does not correctly apply the linearity of the integral, leading to an incorrect result.
qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-object
qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-object
gpt-oss:20b: fail (error) 2026-10-07 — The second line incorrectly substitutes the known integrals: it uses –3·4 instead of –3·(–4) and 0·(–1) for no reason. The correct computation is –[2(–6)–3(–4)]=0.

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 2026-10-07 with SymPy 1.14.0.