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Properties of definite integrals

Problem 4.325 · easy

Suppose \( \displaystyle \int_0^{5} f(x)\,dx = 6 \), \( \displaystyle \int_0^{2} f(x)\,dx = -3 \), \( \displaystyle \int_0^{5} g(x)\,dx = -2 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 6 \). Find \( \displaystyle \int_{2}^{0} \left(f(x) + g(x)\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out; swapping the limits changes the sign.
\[ 1 \left(-6\right) + 0 \left(-2\right) + 1 \cdot 3 = -3 \]
Combine the known values.✓ Proved

Answer \( -3 \)

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution misapplies the given data: it uses the value \int_0^5 f(x)dx = 6 instead of \int_0^2 f(x)dx = -3 when computing the integral from 2 to 0. The
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly calculate the necessary intermediate integral \int_2^5 (f(x) + g(x)) dx using the given bounds (0 to 5 and 0 to 2). It jumps directly to a numerical combination that is not justified by the previous sentence, making the derivation opaque and logically incomplete.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly calculate the necessary intermediate integral \int_2^5 (f(x) + g(x)) dx using the given bounds (0 to 5 and 0 to 2). It jumps directly to a numerical combination that is not justified by the previous sentence, making the derivation opaque and logically incomplete.
gpt-oss:20b: inconclusive 2026-10-04 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution misapplies the given data: it uses the value \int_0^5 f(x)dx = 6 instead of \int_0^2 f(x)dx = -3 when computing the integral from 2 to 0. The
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to compute the necessary sub-integrals (e.g., \int_2^5) from the given data, instead using arbitrary numbers (-6, 0, 3) that do not correspond to the problem statement. The setup is completely disconnected from the provided values.
gpt-oss:20b: inconclusive 2026-10-04 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution misapplies the given integrals: it uses the value \int_0^5 f(x)dx = 6 (or incorrectly -6) instead of the correct \int_0^2 f(x)dx = -3. The co

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 2026-10-04 with SymPy 1.14.0.