Home›Calculus 1›Properties of definite integrals›Problem 4.784

Properties of definite integrals

Problem 4.784 · easy

Suppose \( \displaystyle \int_0^{5} f(x)\,dx = -7 \), \( \displaystyle \int_0^{2} f(x)\,dx = 2 \), \( \displaystyle \int_0^{5} g(x)\,dx = -3 \) and \( \displaystyle \int_0^{2} g(x)\,dx = -3 \). Find \( \displaystyle \int_{0}^{2} \left(3 f(x) + g(x)\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out.
\[ 1 \left(-3\right) + 0 \cdot 2 + 3 \cdot 2 = 3 \]
Combine the known values.✓ Proved

Answer \( 3 \)

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: pass
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to use the given value for \int_0^5 g(x) dx to find \int_2^5 g(x) dx, but more importantly, it does not actually compute \int_2^5 f(x) dx which is not needed, but it incorrectly combines values. Specifically, \int_0^2 (3f+g) = 3\int_0^2 f + \int_0^2 g = 3(2) + (-3) = 3. The equation line `1*(-3) + 0*2 + 3*2 = 3` is numerically correct but the coefficients are unmotivated and confusing. The sentence claims linearity, but the equation mixes terms in a way that doesn't clearly reflect the linearity application (e.g., where did the 0 come from?). It is misleading because it obscures the actual calculation steps.

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.