Properties of definite integrals
Problem 4.491 · easy
Suppose \( \displaystyle \int_0^{6} f(x)\,dx = 5 \), \( \displaystyle \int_0^{1} f(x)\,dx = -7 \), \( \displaystyle \int_0^{6} g(x)\,dx = -1 \) and \( \displaystyle \int_0^{1} g(x)\,dx = 9 \). Find \( \displaystyle \int_{0}^{6} \left(f(x) + g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out.
- \[ 1 \left(-1\right) + 0 \cdot 6 + 1 \cdot 5 = 4 \]Combine the known values.✓ Proved
Answer \( 4 \)
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: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly computes the sum of the integrals. The problem asks for the integral from 0 to 6, which is the sum of the given values 5 and -1, resulting in 4. However, the equation shown '1*(-1) + 0*6 + 1*5 = 4' is nonsensical in this context; it seems to be mixing up values or operations arbitrarily. The correct logic is simply \int_0^6 f(x)dx + \int_0^6 g(x)dx = 5 + (-1) = 4. The provided equation does not represent this logic clearly or correctly.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly computes the sum of the integrals. The problem asks for the integral from 0 to 6, which is the sum of the given values 5 and -1, resulting in 4. However, the equation shown '1*(-1) + 0*6 + 1*5 = 4' is nonsensical in this context; it seems to be mixing up values or operations arbitrarily. The correct logic is simply \int_0^6 f(x)dx + \int_0^6 g(x)dx = 5 + (-1) = 4. The provided equation does not represent this logic clearly or correctly.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: inconclusive 2026-10-06 — 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 2026-10-06 with SymPy 1.14.0.