Home›Calculus 1›Fundamental Theorem of Calculus, Part 1›Problem 4.26

Fundamental Theorem of Calculus, Part 1

Problem 4.26 · medium

Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{x} \sin{\left(t^{2} \right)} \, dt \).

By FTC Part 1 and the chain rule, d/dx ∫ from a to u(x) of g(t) dt = g(u(x)) u'(x).
Reviewed
\[ \frac{d}{d x} x = 1 \]
u'(x).✓ Proved
\[ \sin{\left(x^{2} \right)} \]
g(u(x)) u'(x).Reviewed

Answer \( \sin{\left(x^{2} \right)} \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the integral computed numerically at x = 0.7 ± 10⁻⁶ and differenced agrees

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the Fundamental Theorem of Calculus Part 1 combined with the Chain Rule. The identification of u(x) = x and g(t) = sin(t^2) is correct, leading to the correct derivative sin(x^2).

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the Fundamental Theorem of Calculus Part 1 combined with the Chain Rule. The identification of u(x) = x and g(t) = sin(t^2) is correct, leading to the correct derivative sin(x^2).
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the Fundamental Theorem of Calculus Part 1 combined with the chain rule. The setup is accurate, and the algebraic steps are verified.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the Fundamental Theorem of Calculus Part 1 combined with the chain rule. The identification of u(x)=x, u'(x)=1, and g(u(x))=sin(x^2) is accurate.
gpt-oss:20b: pass 2026-09-26

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_part1, checked 2026-10-13 with SymPy 1.14.0.