Accumulation functions
Problem 4.1202 · easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} 2 t \, dt \) as a function of \( \displaystyle x \).
- \[ \frac{d}{d t} t^{2} = 2 t \]An antiderivative of the integrand is t**2.✓ Proved
- \[ - \left. t^{2} \right|_{\substack{ t=-1 }} + \left. t^{2} \right|_{\substack{ t=x }} = x^{2} - 1 \]F(x) = G(x) − G(a).✓ Proved
- \[ \frac{d}{d x} \left(x^{2} - 1\right) = 2 x \]F′(x) = f(x), as FTC Part 1 says.✓ Proved
- \[ \left. x^{2} - 1 \right|_{\substack{ x=-1 }} = 0 \]And F(-1) = 0.✓ Proved
Answer \( F(x) = x^{2} - 1 \)
✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the Fundamental Theorem of Calculus Part 2 to evaluate the definite integral. The steps are logically sound and the final result is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the Fundamental Theorem of Calculus Part 2 to evaluate the definite integral. The steps are logically sound and the final result is correct.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the Fundamental Theorem of Calculus Part 2 to evaluate the definite integral as a function of x. The steps are logically sound and the final result is correct.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the Fundamental Theorem of Calculus Part 2 to evaluate the definite integral as a function of x. The steps are logically sound and the final answer is correct.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.