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Accumulation functions

Problem 4.1253 · easy

Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} t - 3 \, dt \) as a function of \( \displaystyle x \).

\[ \frac{d}{d t} \left(\frac{t^{2}}{2} - 3 t\right) = t - 3 \]
An antiderivative of the integrand is t**2/2 - 3*t.✓ Proved
\[ - \left. \frac{t^{2}}{2} - 3 t \right|_{\substack{ t=1 }} + \left. \frac{t^{2}}{2} - 3 t \right|_{\substack{ t=x }} = \frac{x^{2}}{2} - 3 x + \frac{5}{2} \]
F(x) = G(x) − G(a).✓ Proved
\[ \frac{d}{d x} \left(\frac{x^{2}}{2} - 3 x + \frac{5}{2}\right) = x - 3 \]
F′(x) = f(x), as FTC Part 1 says.✓ Proved
\[ \left. \frac{x^{2}}{2} - 3 x + \frac{5}{2} \right|_{\substack{ x=1 }} = 0 \]
And F(1) = 0.✓ Proved

Answer \( F(x) = \frac{x^{2}}{2} - 3 x + \frac{5}{2} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ 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	numerical quadrature from a to three values of x

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the Fundamental Theorem of Calculus to evaluate the definite integral and explicitly verifies the result using FTC Part 1 and the initial condition.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the Fundamental Theorem of Calculus to evaluate the definite integral as a function of x. The algebraic steps and verification checks are correct.
gpt-oss:20b: pass 2026-10-16

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/ftc_accumulation, checked 2026-10-19 with SymPy 1.14.0.