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Properties of definite integrals

Problem 4.628 · easy

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

The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 1 to 5 = ∫ from 0 to 5 − ∫ from 0 to 1; swapping the limits changes the sign.
\[ 2 \left(-11\right) + 0 \left(-4\right) - 3 \left(-5\right) = -7 \]
Combine the known values.✓ Proved

Answer \( -7 \)

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 contains a calculation error in the intermediate step. The term '0*(-4)' is incorrect; the coefficient for the f(x) integral should be 2, not 0. Additionally, the values used (-11 and -5) are not explicitly derived from the given integrals in the text, making the logic opaque and the arithmetic wrong.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution contains a calculation error in the intermediate step. The term '0*(-4)' is incorrect; the coefficient for the f(x) integral should be 2, not 0. Additionally, the values used (-11 and -5) are not explicitly derived from the given integrals in the text, making the logic opaque and the arithmetic wrong.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution contains arithmetic errors in calculating the intermediate integrals (e.g., claiming \int_1^5 f(x) dx = -11 instead of 11) and incorrectly includes a term '0*(-4)' which has no basis in the problem statement.
gpt-oss:20b: fail (error) 2026-10-07 — The solution misapplies the limits and signs. ∫₁⁵f(x)dx = 7−(−4)=11, not −11, and ∫₁⁵g(x)dx = 4−(−1)=5, not −5. Moreover, the target integral is from 5 to 1, which introduces an extra minus sign that is not handled. These errors lead to an incorrect conclusion.

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.