Properties of definite integrals
Problem 4.788 · easy
Suppose \( \displaystyle \int_0^{3} f(x)\,dx = -3 \), \( \displaystyle \int_0^{2} f(x)\,dx = -6 \), \( \displaystyle \int_0^{3} g(x)\,dx = -1 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 1 \). Find \( \displaystyle \int_{3}^{2} \left(2 f(x) - 3 g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 2 to 3 = ∫ from 0 to 3 − ∫ from 0 to 2; swapping the limits changes the sign.
- \[ 2 \left(-3\right) - 3 \cdot 2 + 0 \left(-1\right) = -12 \]Combine the known values.✓ Proved
Answer \( -12 \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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: fail (error) — The second sentence incorrectly uses the given integrals: it substitutes 2 for ∫₀²f(x)dx (which is –6) and 0 for ∫₀²g(x)dx (which is 1). The calculation does not follow from the data, so the conclusion is invalid.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — The second sentence incorrectly uses the given integrals: it substitutes 2 for ∫₀²f(x)dx (which is –6) and 0 for ∫₀²g(x)dx (which is 1). The calculation does not follow from the data, so the conclusion is invalid.qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-objectgpt-oss:20b: inconclusive 2026-10-09 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly uses the value of \int_0^2 g(x)\,dx as 2 instead of 1 and omits the necessary sign change when swapping the limits from 3 to 2. Tqwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution incorrectly calculates the integral of g from 2 to 3 as 2, whereas it is -2 (1 - (-1)). This leads to the wrong final answer of -12 instead of -18.
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.