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Fundamental Theorem of Calculus, Part 1

Problem 4.113 · medium

Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{x^{2}} \frac{1}{t^{4} + 1} \, dt \).

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
\[ \frac{d}{d x} x^{2} = 2 x \]
u'(x).✓ Proved
\[ \frac{2 x}{x^{8} + 1} \]
g(u(x)) u'(x).✓ Proved

Answer \( \frac{2 x}{x^{8} + 1} \)

✓ 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 combined with the chain rule. The identification of u(x) and g(t) is implicit but correct, and the algebraic substitution is verified.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the Fundamental Theorem of Calculus combined with the chain rule. The identification of u(x) and g(t) is implicit but correct, and the algebraic substitution is verified.
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 substitution of u(x) = x^2 into g(t) = 1/(t^4 + 1) yields 1/(x^8 + 1), and multiplying by u'(x) = 2x gives the correct result.
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.