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Properties of definite integrals

Problem 4.623 · easy

Suppose \( \displaystyle \int_0^{2} f(x)\,dx = -9 \), \( \displaystyle \int_0^{1} f(x)\,dx = 5 \), \( \displaystyle \int_0^{2} g(x)\,dx = 4 \) and \( \displaystyle \int_0^{1} g(x)\,dx = -1 \). Find \( \displaystyle \int_{1}^{2} \left(2 f(x) - 3 g(x)\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 1 to 2 = ∫ from 0 to 2 − ∫ from 0 to 1.
\[ 2 \left(-14\right) - 3 \cdot 5 + 0 \cdot 1 = -43 \]
Combine the known values.✓ Proved

Answer \( -43 \)

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 explicitly calculate the intermediate integrals \int_1^2 f(x) dx and \int_1^2 g(x) dx, jumping directly to the final arithmetic. While the arithmetic in line 2 is correct based on the implied values (-14 and 5), the sentence in line 1 does not derive these values, making the logical connection between the given data and the final equation opaque and incomplete.

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 explicitly calculate the intermediate integrals \int_1^2 f(x) dx and \int_1^2 g(x) dx, jumping directly to the final arithmetic. While the arithmetic in line 2 is correct based on the implied values (-14 and 5), the sentence in line 1 does not derive these values, making the logical connection between the given data and the final equation opaque and incomplete.
qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly calculates the sub-integrals. It uses -14 for the integral of f from 1 to 2 (which is actually -14) but uses 5 for the inte
gpt-oss:20b: inconclusive 2026-10-07 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution misapplies the given integral values: it uses 2*(-14) and -3*5, which are not the correct integrals of f and g over the specified intervals.

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.