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Properties of definite integrals

Problem 4.629 · easy

Suppose \( \displaystyle \int_0^{6} f(x)\,dx = -5 \), \( \displaystyle \int_0^{2} f(x)\,dx = -5 \), \( \displaystyle \int_0^{6} g(x)\,dx = 3 \) and \( \displaystyle \int_0^{2} g(x)\,dx = -8 \). Find \( \displaystyle \int_{6}^{2} \left(2 f(x) - 3 g(x)\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 2 to 6 = ∫ from 0 to 6 − ∫ from 0 to 2; swapping the limits changes the sign.
\[ 2 \cdot 0 + 0 \left(-4\right) - 3 \left(-11\right) = 33 \]
Combine the known values.✓ Proved

Answer \( 33 \)

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: inconclusive — reviewer returned a non-object

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-object
gpt-oss:20b: inconclusive 2026-10-07 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The first sentence incorrectly states that \(\int_{2}^{6}=\int_{0}^{6}-\int_{0}^{2}\); it should be \(\int_{6}^{2}=-(\int_{0}^{6}-\int_{0}^{2})\). The sec
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution contains a calculation error in the intermediate step. It claims the integral of g from 2 to 6 is -11, but it is actually 11 (3 - (-8)). Consequently, the term -3 * integral(g) should be -33, not +33, making the final answer 0, not 33.

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.