Properties of definite integrals
Problem 4.328 · easy
Suppose \( \displaystyle \int_0^{4} f(x)\,dx = 4 \), \( \displaystyle \int_0^{2} f(x)\,dx = 7 \), \( \displaystyle \int_0^{4} g(x)\,dx = 6 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 8 \). Find \( \displaystyle \int_{0}^{4} \left(2 f(x) - 3 g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out.
- \[ \left(-3\right) 6 + 0 \cdot 4 + 2 \cdot 4 = -10 \]Combine the known values.✓ Proved
Answer \( -10 \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in 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: passqwen3.6:27b-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: pass 2026-10-04qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-04 — The second line incorrectly includes a term "0*4" which is not part of the calculation. The correct computation is 2*∫0^4 f(x)dx + (-3)*∫0^4 g(x)dx = 2*4 + (-3)*6 = -10. The extraneous term indicates a mistake in the reasoning.
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.