Properties of definite integrals
Problem 4.323 · easy
Suppose \( \displaystyle \int_0^{5} f(x)\,dx = 6 \), \( \displaystyle \int_0^{2} f(x)\,dx = -2 \), \( \displaystyle \int_0^{5} g(x)\,dx = 5 \) and \( \displaystyle \int_0^{2} g(x)\,dx = -1 \). Find \( \displaystyle \int_{5}^{2} \left(f(x) + g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 2 to 5 = ∫ from 0 to 5 − ∫ from 0 to 2; swapping the limits changes the sign.
- \[ 1 \left(-8\right) + 1 \left(-6\right) + 0 \left(-3\right) = -14 \]Combine the known values.✓ Proved
Answer \( -14 \)
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 first sentence misstates the relationship between the integral from 5 to 2 and the integrals from 0 to 5 and 0 to 2. It incorrectly claims ∫₂⁵ = ∫₀⁵ – ∫₀², which is true for 2→5 but not for 5→2, and it does not correctly account for the sign change when swapping limits. The subsequent algebraic step is correct, but the setup sentence is flawed.qwen3.6:27b-mlx: fail (error) — The solution contains arithmetic errors in the intermediate steps (claiming -8 and -6 instead of the correct -8 and -6 for the individual integrals, but then summing them incorrectly or using wrong values). Specifically, \int_2^5 f(x) dx = 6 - (-2) = 8, and \int_2^5 g(x) dx = 5 - (-1) = 6. The integral from 5 to 2 is the negative of these, so -8 and -6. The sum is -14. Wait, let me re-read the equation line. The equation line says `1*(-8) + 1*(-6) + 0*(-3) = -14`. This arithmetic is correct (-14). However, the sentence says "\int from 2 to 5 = \int from 0 to 5 - \int from 0 to 2". This is correct for \int_2^5. But the problem asks for \int_5^2. The sentence mentions "swapping the limits changes the sign". The equation line uses -8 and -6. \int_2^5 f = 8. \int_5^2 f = -8. \int_2^5 g = 6. \int_5^2 g = -6. Sum = -14. The logic holds. Why did I think it was wrong? Let's look closer. The sentence is a bit garbled but the logic is: Split sum, swap limits. The equation calculates the values. Is there a hidden error? The term `0*(-3)` is mysterious and unnecessary, suggesting a copy-paste error or confusion with a third function, but it doesn't affect the result. However, the prompt asks to find what is wrong. The sentence is poorly written but technically describes the correct operations. The equation is correct. Is it 'pass'? No, the `0*(-3)` is misleading/confusing. Also, the sentence structure is very poor. But is it an error? The prompt says 'find what is wrong'. The inclusion of `0*(-3)` is a style/misleading issue. But wait, look at the values again. \int_0^5 f = 6, \int_0^2 f = -2. \int_2^5 f = 6 - (-2) = 8. \int_5^2 f = -8. Correct. \int_0^5 g = 5, \int_0^2 g = -1. \int_2^5 g = 5 - (-1) = 6. \int_5^2 g = -6. Correct. Sum = -14. The math is right. The `0*(-3)` is just noise. Is it 'style'? Or 'misleading'? It might confuse a student into thinking there's a third term. I will mark it as misleading due to the extraneous term and poor sentence clarity.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution contains arithmetic errors in the intermediate steps (claiming -8 and -6 instead of the correct -8 and -6 for the individual integrals, but then summing them incorrectly or using wrong values). Specifically, \int_2^5 f(x) dx = 6 - (-2) = 8, and \int_2^5 g(x) dx = 5 - (-1) = 6. The integral from 5 to 2 is the negative of these, so -8 and -6. The sum is -14. Wait, let me re-read the equation line. The equation line says `1*(-8) + 1*(-6) + 0*(-3) = -14`. This arithmetic is correct (-14). However, the sentence says "\int from 2 to 5 = \int from 0 to 5 - \int from 0 to 2". This is correct for \int_2^5. But the problem asks for \int_5^2. The sentence mentions "swapping the limits changes the sign". The equation line uses -8 and -6. \int_2^5 f = 8. \int_5^2 f = -8. \int_2^5 g = 6. \int_5^2 g = -6. Sum = -14. The logic holds. Why did I think it was wrong? Let's look closer. The sentence is a bit garbled but the logic is: Split sum, swap limits. The equation calculates the values. Is there a hidden error? The term `0*(-3)` is mysterious and unnecessary, suggesting a copy-paste error or confusion with a third function, but it doesn't affect the result. However, the prompt asks to find what is wrong. The sentence is poorly written but technically describes the correct operations. The equation is correct. Is it 'pass'? No, the `0*(-3)` is misleading/confusing. Also, the sentence structure is very poor. But is it an error? The prompt says 'find what is wrong'. The inclusion of `0*(-3)` is a style/misleading issue. But wait, look at the values again. \int_0^5 f = 6, \int_0^2 f = -2. \int_2^5 f = 6 - (-2) = 8. \int_5^2 f = -8. Correct. \int_0^5 g = 5, \int_0^2 g = -1. \int_2^5 g = 5 - (-1) = 6. \int_5^2 g = -6. Correct. Sum = -14. The math is right. The `0*(-3)` is just noise. Is it 'style'? Or 'misleading'? It might confuse a student into thinking there's a third term. I will mark it as misleading due to the extraneous term and poor sentence clarity.gpt-oss:20b: fail (error) 2026-10-04 — The first sentence misstates the relationship between the integral from 5 to 2 and the integrals from 0 to 5 and 0 to 2. It incorrectly claims ∫₂⁵ = ∫₀⁵ – ∫₀², which is true for 2→5 but not for 5→2, and it does not correctly account for the sign change when swapping limits. The subsequent algebraic step is correct, but the setup sentence is flawed.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly calculate the intermediate integrals \int_2^5 f(x) dx and \int_2^5 g(x) dx, jumping directly to a confusing arithmetic expression that does not clearly follow from the stated properties. Specifically, it is unclear how the values -8, -6, and -3 are derived from the given data without showing the subtraction steps (e.g., 6 - (-2) = 8).gpt-oss:20b: fail (error) 2026-10-04 — The first sentence incorrectly sets up the integral limits: the problem asks for ∫₅²(f+g), but the solution rewrites ∫₂⁵. Moreover, the algebraic step uses –8 and –6 instead of the correct 8 and 6, leading to an incorrect sign. The setup and calculation are both wrong.
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 2026-10-04 with SymPy 1.14.0.