Properties of definite integrals
Problem 4.625 · easy
Suppose \( \displaystyle \int_0^{6} f(x)\,dx = -5 \), \( \displaystyle \int_0^{2} f(x)\,dx = -4 \), \( \displaystyle \int_0^{6} g(x)\,dx = -3 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 2 \). Find \( \displaystyle \int_{6}^{0} \left(f(x) + g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; swapping the limits changes the sign.
- \[ 0 \left(-6\right) + 1 \cdot 3 + 1 \cdot 5 = 8 \]Combine the known values.✓ Proved
Answer \( 8 \)
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 an incorrect algebraic expression (0*(-6)+1*3+1*5=8) that does not correctly combine the given integrals. The correct computation should be -[(-5)+(-3)]=8. The sentence is fine, but the equation is mathematically invalid.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — The solution’s second line contains an incorrect algebraic expression (0*(-6)+1*3+1*5=8) that does not correctly combine the given integrals. The correct computation should be -[(-5)+(-3)]=8. The sentence is fine, but the equation is mathematically invalid.qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-objectqwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-objectgpt-oss:20b: inconclusive 2026-10-07 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution’s second line uses an incorrect algebraic expression (0*(-6)+1*3+1*5) that does not correctly combine the given integrals. The correct comput
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 2026-10-07 with SymPy 1.14.0.