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Properties of definite integrals

Problem 4.326 · easy

Suppose \( \displaystyle \int_0^{3} f(x)\,dx = -4 \), \( \displaystyle \int_0^{2} f(x)\,dx = -2 \), \( \displaystyle \int_0^{3} g(x)\,dx = -3 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 8 \). Find \( \displaystyle \int_{2}^{0} \left(f(x) + 2\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out; swapping the limits changes the sign.
\[ 2 \left(-2\right) + 0 \left(-8\right) + 1 \cdot 2 = -2 \]
Combine the known values.✓ Proved

Answer \( -2 \)

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: fail (error) — The solution misapplies the sign change for the integral of f and incorrectly uses the constant term. It produces the correct numeric answer by accident, but the reasoning is incorrect and would mislead a student.
qwen3.6:27b-mlx: fail (error) — The solution fails to split the integral of the constant term correctly, ignoring the interval length (2-0=2). It also incorrectly uses the value of g(x) which is irrelevant to the problem.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to split the integral of the constant term correctly, ignoring the interval length (2-0=2). It also incorrectly uses the value of g(x) which is irrelevant to the problem.
gpt-oss:20b: fail (error) 2026-10-04 — The solution misapplies the sign change for the integral of f and incorrectly uses the constant term. It produces the correct numeric answer by accident, but the reasoning is incorrect and would mislead a student.
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer returned a non-object
gpt-oss:20b: fail (error) 2026-10-04 — The solution incorrectly incorporates the function g and its integrals, which are irrelevant to the requested integral. The calculation 2*(-2)+0*(-8)+1*2 is not justified and misuses the given data.

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.