Properties of definite integrals
Problem 4.791 · easy
Suppose \( \displaystyle \int_0^{5} f(x)\,dx = -5 \), \( \displaystyle \int_0^{2} f(x)\,dx = -1 \), \( \displaystyle \int_0^{5} g(x)\,dx = 7 \) and \( \displaystyle \int_0^{2} g(x)\,dx = -6 \). Find \( \displaystyle \int_{0}^{2} \left(2 f(x) - 3 g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out.
- \[ 2 \left(-1\right) + 0 \cdot 2 - 3 \left(-6\right) = 16 \]Combine the known values.✓ Proved
Answer \( 16 \)
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 2026-10-09qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution incorrectly assumes the integral of f(x) from 0 to 2 is -1 without justifying it, but more critically, it introduces a term '0*2' which has no basis in the problem statement or the linearity properties. The correct calculation should be 2*(-1) - 3*(-6) = 16, but the presence of the '0*2' term suggests a fundamental misunderstanding or a hallucinated step, making the reasoning flawed even if the final number happens to be correct by coincidence (if we ignore the extra term). Actually, looking closer: 2*(-1) - 3*(-6) = -2 + 18 = 16. The equation line says `2*(-1) + 0*2 - 3*(-6) = 16`. The `0*2` is extraneous and unjustified. While the final answer is correct, the derivation contains a nonsensical term.
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.