Properties of definite integrals
Problem 4.631 · easy
Suppose \( \displaystyle \int_0^{5} f(x)\,dx = 5 \), \( \displaystyle \int_0^{4} f(x)\,dx = -4 \), \( \displaystyle \int_0^{5} g(x)\,dx = 9 \) and \( \displaystyle \int_0^{4} g(x)\,dx = -8 \). Find \( \displaystyle \int_{4}^{5} \left(2 f(x) - 3 g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 4 to 5 = ∫ from 0 to 5 − ∫ from 0 to 4.
- \[ \left(-3\right) 17 + 0 \cdot 1 + 2 \cdot 9 = -33 \]Combine the known values.✓ Proved
Answer \( -33 \)
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 2026-10-07qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-objectqwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly calculates the integral of f from 4 to 5 as 0 (likely using 5 - 5 instead of 5 - (-4) = 9). The correct value for int_4^5 fgpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.