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Fundamental Theorem of Calculus, Part 1

Problem 4.62 · medium

Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{3 x} \sin{\left(t^{2} \right)} \, dt \).

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
\[ \frac{d}{d x} 3 x = 3 \]
u'(x).✓ Proved
\[ 3 \sin{\left(9 x^{2} \right)} \]
g(u(x)) u'(x).Reviewed

Answer \( 3 \sin{\left(9 x^{2} \right)} \)

✓ 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 substitution of the upper limit into the integrand and multiplication by the derivative of the limit are performed correctly.

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 substitution of the upper limit into the integrand and multiplication by the derivative of the limit are performed correctly.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the Fundamental Theorem of Calculus combined with the Chain Rule. The substitution u(x) = 3x and g(t) = sin(t^2) leads to g(u(x))u'(x) = sin((3x)^2) * 3 = 3sin(9x^2), which matches the stated answer.
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 substitution of u(x) = 3x into g(t) = sin(t^2) yields sin((3x)^2) = sin(9x^2), and multiplying by u'(x) = 3 gives the correct result.
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.