Fundamental Theorem of Calculus, Part 1
Problem 4.99 · medium
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{x^{3}} \frac{1}{t^{4} + 1} \, dt \).
- 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
- \[ \frac{d}{d x} x^{3} = 3 x^{2} \]u'(x).✓ Proved
- \[ \frac{3 x^{2}}{x^{12} + 1} \]g(u(x)) u'(x).✓ Proved
Answer \( \frac{3 x^{2}}{x^{12} + 1} \)
✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the Fundamental Theorem of Calculus Part 1 combined with the Chain Rule. The identification of u(x) and g(t) is implicit but correct, and the final algebraic substitution is accurate.
Every verdict on record (4)
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) and g(t) is implicit but correct, and the final algebraic substitution is accurate.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-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) and g(t) is correct, and the final algebraic simplification 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 2026-09-26 with SymPy 1.14.0.