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Fundamental Theorem of Calculus, Part 1

Problem 4.117 · medium

Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{1}^{x^{3}} t^{3} \cos{\left(t \right)} \, dt \).

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
\[ \frac{d}{d x} x^{3} = 3 x^{2} \]
u'(x).✓ Proved
\[ 3 x^{11} \cos{\left(x^{3} \right)} \]
g(u(x)) u'(x).✓ Proved

Answer \( 3 x^{11} \cos{\left(x^{3} \right)} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

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	✓ 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	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 intermediate steps and final result are algebraically correct.

Every verdict on record (4)

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 intermediate steps and final result are algebraically correct.
gpt-oss:20b: pass 2026-09-27
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 identification of u(x) = x^3, u'(x) = 3x^2, and g(t) = t^3 cos(t) leads to the correct final expression 3x^11 cos(x^3).
gpt-oss:20b: pass 2026-09-27

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 2026-09-27 with SymPy 1.14.0.