Properties of definite integrals
Problem 4.496 · easy
Suppose \( \displaystyle \int_0^{6} f(x)\,dx = 8 \), \( \displaystyle \int_0^{3} f(x)\,dx = -2 \), \( \displaystyle \int_0^{6} g(x)\,dx = 9 \) and \( \displaystyle \int_0^{3} g(x)\,dx = -5 \). Find \( \displaystyle \int_{0}^{3} \left(3 f(x) + g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out.
- \[ 3 \left(-2\right) + 1 \left(-5\right) + 0 \cdot 3 = -11 \]Combine the known values.✓ Proved
Answer \( -11 \)
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: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly assumes that the integral from 0 to 3 is the only component needed, ignoring that the problem provides integrals up to 6 which might be a distractor, but more critically, the equation line is syntactically nonsensical ('+ 0*3') and does not follow from the sentence. The sentence states the linearity property, but the equation attempts to compute the result by just plugging in the 0-to-3 values. While the final number -11 is correct (3*(-2) + (-5) = -11), the equation line is malformed and the '0*3' term is unexplained and incorrect notation for the calculation steps. A proper solution would explicitly write \int_0^3 (3f+g) = 3\int_0^3 f + \int_0^3 g = 3(-2) + (-5) = -11. The provided equation is garbage.
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.