Properties of definite integrals
Problem 4.499 · easy
Suppose \( \displaystyle \int_0^{6} f(x)\,dx = -5 \), \( \displaystyle \int_0^{1} f(x)\,dx = -4 \), \( \displaystyle \int_0^{6} g(x)\,dx = 3 \) and \( \displaystyle \int_0^{1} g(x)\,dx = -2 \). Find \( \displaystyle \int_{0}^{1} \left(f(x) + 3\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out.
- \[ 1 \left(-4\right) + 0 \left(-2\right) + 3 \cdot 1 = -1 \]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. 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: pass — The solution correctly applies linearity of the integral and uses the given value ∫₀¹f(x)dx = –4 together with ∫₀¹3dx = 3 to obtain –1. The inclusion of the term 0·(–2) is unnecessary but does not affect the result. No mathematical error is present.qwen3.6:27b-mlx: fail (error) — The solution incorrectly includes the integral of g(x) in the calculation for an expression that only involves f(x). The term 0*(-2) suggests a confusion with a different problem or a hallucinated dependency on g(x).
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06 — The solution correctly applies linearity of the integral and uses the given value ∫₀¹f(x)dx = –4 together with ∫₀¹3dx = 3 to obtain –1. The inclusion of the term 0·(–2) is unnecessary but does not affect the result. No mathematical error is present.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly includes the integral of g(x) in the calculation for an expression that only involves f(x). The term 0*(-2) suggests a confusion with a different problem or a hallucinated dependency on g(x).gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly includes terms involving g(x) (specifically 0*(-2)), which are irrelevant to the integral of f(x) + 3. The calculation should simply be \int_0^1 f(x)dx + \int_0^1 3 dx = -4 + 3 = -1, without referencing g(x).
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.