Properties of definite integrals
Problem 4.331 · easy
Suppose \( \displaystyle \int_0^{6} f(x)\,dx = -1 \), \( \displaystyle \int_0^{1} f(x)\,dx = 9 \), \( \displaystyle \int_0^{6} g(x)\,dx = -9 \) and \( \displaystyle \int_0^{1} g(x)\,dx = 7 \). Find \( \displaystyle \int_{6}^{1} \left(f(x) + 3\right) dx \).
- 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; swapping the limits changes the sign.
- \[ 3 \left(-5\right) + 0 \cdot 16 + 1 \cdot 10 = -5 \]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 first sentence incorrectly states the bounds (it uses 1 to 6 instead of 6 to 1) and does not justify the use of the given integrals. The second line is a nonsensical algebraic combination that does not follow from any valid manipulation of the data. The reasoning is incorrect and would mislead a student.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-04 — The solution’s first sentence incorrectly states the bounds (it uses 1 to 6 instead of 6 to 1) and does not justify the use of the given integrals. The second line is a nonsensical algebraic combination that does not follow from any valid manipulation of the data. The reasoning is incorrect and would mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to use the provided values for f(x) and g(x), instead using arbitrary numbers (10, 16) that do not correspond to the problem statement. Additionally, the integral of the constant 3 is calculated incorrectly or omitted in the final combination.gpt-oss:20b: fail (error) 2026-10-04 — The solution’s first sentence incorrectly states the bounds (it uses 1 to 6 instead of 6 to 1) and does not correctly apply the linearity of the integral or the constant factor. The second line is a meaningless algebraic expression that does not follow from any preceding steps. The reasoning is therefore incorrect and would mislead a student.
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.