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Properties of definite integrals

Problem 4.943 · easy

Suppose \( \displaystyle \int_0^{4} f(x)\,dx = 2 \), \( \displaystyle \int_0^{3} f(x)\,dx = -7 \), \( \displaystyle \int_0^{4} g(x)\,dx = -1 \) and \( \displaystyle \int_0^{3} g(x)\,dx = 5 \). Find \( \displaystyle \int_{0}^{4} \left(3 f(x) + g(x)\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out.
\[ 1 \left(-1\right) + 0 \cdot 4 + 3 \cdot 2 = 5 \]
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: fail (misleading) — The second line mixes unrelated terms (0*4) and does not correctly express the application of linearity; it could mislead a student about how to combine the given integrals.
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (4)

gpt-oss:20b: fail (misleading) 2026-10-12 — The second line mixes unrelated terms (0*4) and does not correctly express the application of linearity; it could mislead a student about how to combine the given integrals.
qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to use the given values for the integrals from 0 to 3, which are necessary to determine the integral from 3 to 4. It incorrectly assumes the integral from 0 to 4 can be computed directly without splitting the interval, leading to an incorrect application of linearity.

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 2026-10-12 with SymPy 1.14.0.