Properties of definite integrals
Problem 4.786 · easy
Suppose \( \displaystyle \int_0^{6} f(x)\,dx = 6 \), \( \displaystyle \int_0^{5} f(x)\,dx = 7 \), \( \displaystyle \int_0^{6} g(x)\,dx = 7 \) and \( \displaystyle \int_0^{5} g(x)\,dx = 8 \). Find \( \displaystyle \int_{5}^{6} \left(f(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 5 to 6 = ∫ from 0 to 6 − ∫ from 0 to 5.
- \[ 1 \left(-1\right) + 0 \left(-1\right) + 0 \cdot 1 = -1 \]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 second sentence attempts to combine known values with a nonsensical equation 1*(-1)+0*(-1)+0*1 = -1, which does not follow from the given data or any valid algebraic manipulation. The correct reasoning is simply ∫₅⁶ f(x)dx = ∫₀⁶ f(x)dx – ∫₀⁵ f(x)dx = 6 – 7 = –1.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly calculate the value of the integral using the given data (6 - 7 = -1) before stating the final answer. It jumps to a nonsensical arithmetic expression '1*(-1) + 0*(-1) + 0*1' that does not correspond to the problem statement, making the derivation unclear and potentially confusing for a student.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — The second sentence attempts to combine known values with a nonsensical equation 1*(-1)+0*(-1)+0*1 = -1, which does not follow from the given data or any valid algebraic manipulation. The correct reasoning is simply ∫₅⁶ f(x)dx = ∫₀⁶ f(x)dx – ∫₀⁵ f(x)dx = 6 – 7 = –1.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explicitly calculate the value of the integral using the given data (6 - 7 = -1) before stating the final answer. It jumps to a nonsensical arithmetic expression '1*(-1) + 0*(-1) + 0*1' that does not correspond to the problem statement, making the derivation unclear and potentially confusing for a student.gpt-oss:20b: fail (misleading) 2026-10-09 — The second sentence is unclear and incorrectly describes how the known integrals are combined. It does not correctly state that ∫₅⁶ f = ∫₀⁶ f – ∫₀⁵ f = 6 – 7 = –1, and the algebra shown is nonsensical.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explicitly state the calculation \int_5^6 f(x) dx = \int_0^6 f(x) dx - \int_0^5 f(x) dx = 6 - 7 = -1. Instead, it presents an opaque and incorrect arithmetic expression '1*(-1) + 0*(-1) + 0*1' that does not clearly derive from the given integrals, making the logic unintelligible and the derivation invalid.
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.