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Properties of definite integrals

Problem 4.951 · easy

Suppose \( \displaystyle \int_0^{5} f(x)\,dx = -6 \), \( \displaystyle \int_0^{4} f(x)\,dx = -3 \), \( \displaystyle \int_0^{5} g(x)\,dx = -4 \) and \( \displaystyle \int_0^{4} g(x)\,dx = 7 \). Find \( \displaystyle \int_{0}^{4} \left(f(x) + 3\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out.
\[ 1 \left(-3\right) + 0 \cdot 7 + 3 \cdot 4 = 9 \]
Combine the known values.✓ Proved

Answer \( 9 \)

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 incorrectly includes the integral of g(x) (the term 0*7) in the calculation for an integral involving only f(x). The term 0*7 is irrelevant and suggests a confusion with a different problem or a copy-paste error.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly includes the integral of g(x) (the term 0*7) in the calculation for an integral involving only f(x). The term 0*7 is irrelevant and suggests a confusion with a different problem or a copy-paste error.
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly includes the integral of g(x) in the calculation for an expression that only involves f(x). The term '0*7' suggests a confusion with a different problem or a hallucination of terms not present in the integrand f(x) + 3.

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.