Properties of definite integrals
Problem 4.785 · easy
Suppose \( \displaystyle \int_0^{4} f(x)\,dx = -1 \), \( \displaystyle \int_0^{1} f(x)\,dx = 6 \), \( \displaystyle \int_0^{4} g(x)\,dx = 3 \) and \( \displaystyle \int_0^{1} g(x)\,dx = 7 \). Find \( \displaystyle \int_{1}^{4} \left(- f(x) + 2 g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 1 to 4 = ∫ from 0 to 4 − ∫ from 0 to 1.
- \[ 2 \left(-4\right) + 0 \cdot 3 - 1 \left(-7\right) = -1 \]Combine the known values.✓ Proved
Answer \( -1 \)
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’s algebraic step misuses the given integrals: it substitutes –4 for ∫₀⁴f and 0 for the coefficient of ∫₀⁴g, leading to an incorrect computation. The correct evaluation uses the given values –1, 6, 3, and 7 as shown in the reasoning above.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly calculate the intermediate integrals \int_1^4 f(x) dx and \int_1^4 g(x) dx using the additive property of integrals, jumping directly to a numerical combination that is not clearly derived from the given data. Specifically, it does not show that \int_1^4 f(x) dx = -7 and \int_1^4 g(x) dx = -4, which are necessary steps to justify the final calculation.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — The solution’s algebraic step misuses the given integrals: it substitutes –4 for ∫₀⁴f and 0 for the coefficient of ∫₀⁴g, leading to an incorrect computation. The correct evaluation uses the given values –1, 6, 3, and 7 as shown in the reasoning above.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explicitly calculate the intermediate integrals \int_1^4 f(x) dx and \int_1^4 g(x) dx using the additive property of integrals, jumping directly to a numerical combination that is not clearly derived from the given data. Specifically, it does not show that \int_1^4 f(x) dx = -7 and \int_1^4 g(x) dx = -4, which are necessary steps to justify the final calculation.gpt-oss:20b: fail (error) 2026-10-09 — The second sentence uses incorrect values (e.g., 2*(-4) and 0*3) and does not correctly apply the given integrals to compute ∫₁⁴(-f+2g).qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-object
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 2026-10-09 with SymPy 1.14.0.