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Properties of definite integrals

Problem 4.632 · easy

Suppose \( \displaystyle \int_0^{3} f(x)\,dx = -4 \), \( \displaystyle \int_0^{1} f(x)\,dx = 9 \), \( \displaystyle \int_0^{3} g(x)\,dx = 8 \) and \( \displaystyle \int_0^{1} g(x)\,dx = 8 \). Find \( \displaystyle \int_{1}^{0} \left(f(x) + 5\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out; swapping the limits changes the sign.
\[ 1 \left(-9\right) + 5 \left(-1\right) + 0 \left(-8\right) = -14 \]
Combine the known values.✓ Proved

Answer \( -14 \)

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: fail (error) — The solution fails to calculate the necessary integral of f(x) from 1 to 3 using the given data, and incorrectly introduces g(x) (which is irrelevant to the target integral) with arbitrary coefficients. The target integral depends only on f(x), not g(x).

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to calculate the necessary integral of f(x) from 1 to 3 using the given data, and incorrectly introduces g(x) (which is irrelevant to the target integral) with arbitrary coefficients. The target integral depends only on f(x), not g(x).
qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-07

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.