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Properties of definite integrals

Problem 4.949 · easy

Suppose \( \displaystyle \int_0^{3} f(x)\,dx = 7 \), \( \displaystyle \int_0^{1} f(x)\,dx = -3 \), \( \displaystyle \int_0^{3} g(x)\,dx = -8 \) and \( \displaystyle \int_0^{1} g(x)\,dx = 9 \). Find \( \displaystyle \int_{0}^{1} \left(f(x)\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out.
Reviewed
\[ 1 \left(-3\right) + 0 \cdot 9 + 0 \cdot 1 = -3 \]
Combine the known values.✓ Proved

Answer \( -3 \)

Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies that the value of the integral is given directly in the problem statement. The arithmetic in the equation line is a valid, albeit unnecessary, verification of the given value.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly identifies that the value of the integral is given directly in the problem statement. The arithmetic in the equation line is a valid, albeit unnecessary, verification of the given value.
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer returned a non-object

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.