Properties of definite integrals
Problem 4.787 · easy
Suppose \( \displaystyle \int_0^{5} f(x)\,dx = 7 \), \( \displaystyle \int_0^{1} f(x)\,dx = -9 \), \( \displaystyle \int_0^{5} g(x)\,dx = -7 \) and \( \displaystyle \int_0^{1} g(x)\,dx = -2 \). Find \( \displaystyle \int_{5}^{1} \left(3 f(x) + g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 1 to 5 = ∫ from 0 to 5 − ∫ from 0 to 1; swapping the limits changes the sign.
- \[ 3 \left(-16\right) + 0 \left(-4\right) + 1 \cdot 5 = -43 \]Combine the known values.✓ Proved
Answer \( -43 \)
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 uses incorrect values: it writes 0*(-4) instead of the correct contribution from g. The correct computation is 3*(-16)+5=-43. The solution misstates the intermediate numbers, which would mislead a student.qwen3.6:27b-mlx: fail (error) — The solution contains a severe arithmetic error: it claims 3*(-16) + 0*(-4) + 1*5 = -43, but 3*(-16) + 5 is actually -43 only if the second term is 0, yet the calculation of the integrals themselves is also garbled. Specifically, ∫_1^5 f(x)dx = 7 - (-9) = 16, so 3*16 = 48. ∫_1^5 g(x)dx = -7 - (-2) = -5. The integral is ∫_5^1 (...) = - ∫_1^5 (...) = -(3*16 + (-5)) = -(48 - 5) = -43. Wait, let me re-calculate. ∫_0^5 f = 7, ∫_0^1 f = -9 => ∫_1^5 f = 16. ∫_0^5 g = -7, ∫_0^1 g = -2 => ∫_1^5 g = -5. ∫_1^5 (3f+g) = 3(16) + (-5) = 48 - 5 = 43. The problem asks for ∫_5^1, which is -43. The final answer -43 is correct. However, the equation line says 3*(-16) + 0*(-4) + 1*5. Where did -16 come from? ∫_1^5 f is 16, not -16. If they meant ∫_5^1 f, that is -16. Then 3*(-16) = -48. ∫_5^1 g is 5. So -48 + 5 = -43. The equation line has '0*(-4)' which is nonsensical garbage, and '1*5' which corresponds to the g term. But the coefficient for g is 1, and the integral is 5. So 3*(-16) + 1*5 = -43. The '0*(-4)' term is completely unexplained and incorrect syntax for the problem. It looks like a hallucinated term. While the final number is right, the intermediate equation is mathematically incoherent and misleading.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — The second sentence uses incorrect values: it writes 0*(-4) instead of the correct contribution from g. The correct computation is 3*(-16)+5=-43. The solution misstates the intermediate numbers, which would mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution contains a severe arithmetic error: it claims 3*(-16) + 0*(-4) + 1*5 = -43, but 3*(-16) + 5 is actually -43 only if the second term is 0, yet the calculation of the integrals themselves is also garbled. Specifically, ∫_1^5 f(x)dx = 7 - (-9) = 16, so 3*16 = 48. ∫_1^5 g(x)dx = -7 - (-2) = -5. The integral is ∫_5^1 (...) = - ∫_1^5 (...) = -(3*16 + (-5)) = -(48 - 5) = -43. Wait, let me re-calculate. ∫_0^5 f = 7, ∫_0^1 f = -9 => ∫_1^5 f = 16. ∫_0^5 g = -7, ∫_0^1 g = -2 => ∫_1^5 g = -5. ∫_1^5 (3f+g) = 3(16) + (-5) = 48 - 5 = 43. The problem asks for ∫_5^1, which is -43. The final answer -43 is correct. However, the equation line says 3*(-16) + 0*(-4) + 1*5. Where did -16 come from? ∫_1^5 f is 16, not -16. If they meant ∫_5^1 f, that is -16. Then 3*(-16) = -48. ∫_5^1 g is 5. So -48 + 5 = -43. The equation line has '0*(-4)' which is nonsensical garbage, and '1*5' which corresponds to the g term. But the coefficient for g is 1, and the integral is 5. So 3*(-16) + 1*5 = -43. The '0*(-4)' term is completely unexplained and incorrect syntax for the problem. It looks like a hallucinated term. While the final number is right, the intermediate equation is mathematically incoherent and misleading.gpt-oss:20b: fail (misleading) 2026-10-09 — The solution’s second sentence cites incorrect intermediate values (e.g., 0*(-4) and 3*(-16)) yet arrives at the correct final answer. A student would be misled into believing those numbers are derived from the given data, which is false.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution contains arithmetic errors and incorrect intermediate values. Specifically, it uses -16 for the integral of f from 1 to 5 (should be 16) and 5 for the integral of g from 1 to 5 (should be -5). The correct calculation is 3*(16) + (-5) = 43, but the limits are swapped in the problem statement (5 to 1), so the final answer should be -43. However, the line shown calculates -43 using wrong intermediate numbers (-16 and 5) that happen to yield the correct final result by coincidence of sign errors, making the reasoning fundamentally flawed.
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.