Properties of definite integrals
Problem 4.789 · easy
Suppose \( \displaystyle \int_0^{3} f(x)\,dx = 5 \), \( \displaystyle \int_0^{1} f(x)\,dx = -8 \), \( \displaystyle \int_0^{3} g(x)\,dx = 9 \) and \( \displaystyle \int_0^{1} g(x)\,dx = 4 \). Find \( \displaystyle \int_{1}^{3} \left(g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 1 to 3 = ∫ from 0 to 3 − ∫ from 0 to 1.
- \[ 0 \cdot 13 + 0 \cdot 2 + 1 \cdot 5 = 5 \]Combine the known values.✓ Proved
Answer \( 5 \)
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 solution’s second line is nonsensical: it attempts to combine the known integrals with a meaningless expression 0*13 + 0*2 + 1*5. The correct computation is ∫₁³g(x)dx = ∫₀³g(x)dx – ∫₀¹g(x)dx = 9 – 4 = 5.qwen3.6:27b-mlx: fail (error) — The solution fails to use the given values for g(x) and incorrectly applies the linearity property to a single integral without justification. The correct calculation requires subtracting the integral of g from 0 to 1 from the integral of g from 0 to 3.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — The solution’s second line is nonsensical: it attempts to combine the known integrals with a meaningless expression 0*13 + 0*2 + 1*5. The correct computation is ∫₁³g(x)dx = ∫₀³g(x)dx – ∫₀¹g(x)dx = 9 – 4 = 5.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to use the given values for g(x) and incorrectly applies the linearity property to a single integral without justification. The correct calculation requires subtracting the integral of g from 0 to 1 from the integral of g from 0 to 3.gpt-oss:20b: fail (error) 2026-10-09 — The second sentence presents an ill‑formed equation (0*13 + 0*2 + 1*5 = 5) that does not follow from any stated facts. It incorrectly attempts to combine the given integrals and introduces meaningless coefficients, thereby misleading the reader about how the result 5 is obtained. The correct reasoning is ∫₁³g = ∫₀³g – ∫₀¹g = 9 – 4 = 5.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to use the provided value for the integral of g(x) from 0 to 3, instead incorrectly combining unrelated values (likely from f(x) or arbitrary constants) to arrive at the answer. The correct calculation should be \int_0^3 g(x)dx - \int_0^1 g(x)dx = 9 - 4 = 5, but the reasoning shown is nonsensical.
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.