Properties of definite integrals
Problem 4.947 · easy
Suppose \( \displaystyle \int_0^{6} f(x)\,dx = -7 \), \( \displaystyle \int_0^{1} f(x)\,dx = -8 \), \( \displaystyle \int_0^{6} g(x)\,dx = -4 \) and \( \displaystyle \int_0^{1} g(x)\,dx = -4 \). Find \( \displaystyle \int_{1}^{6} \left(3 f(x) + g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 1 to 6 = ∫ from 0 to 6 − ∫ from 0 to 1.
- \[ 1 \cdot 0 + 0 \cdot 5 + 3 \cdot 1 = 3 \]Combine the known values.✓ Proved
Answer \( 3 \)
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 contains a nonsensical calculation. It does not correctly apply the linearity of the integral or the given data to compute ∫₁⁶(3f+g). The correct computation is 3(∫₀⁶f−∫₀¹f)+ (∫₀⁶g−∫₀¹g)=3(−7+8)+0=3, but the presented equation 1*0+0*5+3*1=3 is incorrect and misleading.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly calculate the intermediate integrals \int_1^6 f(x) dx and \int_1^6 g(x) dx using the given values before combining them. While the final arithmetic is correct, the step 'Combine the known values' skips the crucial application of the additive property of integrals (\int_0^6 - \int_0^1) to the specific functions f and g, making the derivation opaque and potentially misleading about how the coefficients 3 and 1 are applied.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — The solution’s second line contains a nonsensical calculation. It does not correctly apply the linearity of the integral or the given data to compute ∫₁⁶(3f+g). The correct computation is 3(∫₀⁶f−∫₀¹f)+ (∫₀⁶g−∫₀¹g)=3(−7+8)+0=3, but the presented equation 1*0+0*5+3*1=3 is incorrect and misleading.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to explicitly calculate the intermediate integrals \int_1^6 f(x) dx and \int_1^6 g(x) dx using the given values before combining them. While the final arithmetic is correct, the step 'Combine the known values' skips the crucial application of the additive property of integrals (\int_0^6 - \int_0^1) to the specific functions f and g, making the derivation opaque and potentially misleading about how the coefficients 3 and 1 are applied.gpt-oss:20b: fail (error) 2026-10-12 — The second line attempts to combine the given integrals but uses an incorrect expression (1*0 + 0*5 + 3*1 = 3). It does not correctly apply the linearity of the integral or the subtraction of the 0–1 intervals. The correct computation is 3(∫0^6 f – ∫0^1 f) + (∫0^6 g – ∫0^1 g) = 3(–7 + 8) + (–4 + 4) = 3.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to perform the actual calculation using the given integral values. It provides a nonsensical arithmetic expression (1*0 + 0*5 + 3*1) that does not correspond to the problem data or the correct application of linearity and additivity properties.
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 2026-10-12 with SymPy 1.14.0.