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Fundamental Theorem of Calculus, Part 1

Problem 4.28 · medium

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

By FTC Part 1 and the chain rule, d/dx ∫ from a to u(x) of g(t) dt = g(u(x)) u'(x).
\[ \frac{d}{d x} \sqrt{x} = \frac{1}{2 \sqrt{x}} \]
u'(x).✓ Proved
\[ \frac{\sqrt{x + 4}}{2 \sqrt{x}} \]
g(u(x)) u'(x).

Answer \( \frac{\sqrt{x + 4}}{2 \sqrt{x}} \)

Lines: 1 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	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: fail (error) — The solution incorrectly substitutes u(x) into g(t). Since g(t) = sqrt(t^2 + 4), g(u(x)) should be sqrt((sqrt(x))^2 + 4) = sqrt(x + 4). However, the final answer provided is sqrt(x + 4)/(2*sqrt(x)), which implies g(u(x)) was calculated as sqrt(x + 4). Wait, let me re-read carefully. g(t) = sqrt(t^2 + 4). u(x) = sqrt(x). g(u(x)) = sqrt((sqrt(x))^2 + 4) = sqrt(x + 4). u'(x) = 1/(2*sqrt(x)). Product is sqrt(x + 4)/(2*sqrt(x)). The stated answer matches this. The equation line 3 shows 'sqrt(x + 4)/(2*sqrt(x))' on both sides, implying it's an identity check or just stating the result. The sentence in step 1 is correct. The derivation is correct. Why did I think it was wrong? Let me re-read the prompt's stated answer: 'sqrt(x + 4)/(2*sqrt(x))'. My calculation: sqrt(x+4)/(2*sqrt(x)). They match. Is there a subtle error? No. The solution is correct. Wait, looking at line 3: 'sqrt(x + 4)/(2*sqrt(x)), sqrt(x + 4)/(2*sqrt(x))'. This is just confirming the result. The logic holds. Verdict: pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly substitutes u(x) into g(t). Since g(t) = sqrt(t^2 + 4), g(u(x)) should be sqrt((sqrt(x))^2 + 4) = sqrt(x + 4). However, the final answer provided is sqrt(x + 4)/(2*sqrt(x)), which implies g(u(x)) was calculated as sqrt(x + 4). Wait, let me re-read carefully. g(t) = sqrt(t^2 + 4). u(x) = sqrt(x). g(u(x)) = sqrt((sqrt(x))^2 + 4) = sqrt(x + 4). u'(x) = 1/(2*sqrt(x)). Product is sqrt(x + 4)/(2*sqrt(x)). The stated answer matches this. The equation line 3 shows 'sqrt(x + 4)/(2*sqrt(x))' on both sides, implying it's an identity check or just stating the result. The sentence in step 1 is correct. The derivation is correct. Why did I think it was wrong? Let me re-read the prompt's stated answer: 'sqrt(x + 4)/(2*sqrt(x))'. My calculation: sqrt(x+4)/(2*sqrt(x)). They match. Is there a subtle error? No. The solution is correct. Wait, looking at line 3: 'sqrt(x + 4)/(2*sqrt(x)), sqrt(x + 4)/(2*sqrt(x))'. This is just confirming the result. The logic holds. Verdict: pass.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly substitutes u(x) into g(t). Since g(t) = sqrt(t^2 + 4), g(u(x)) should be sqrt((sqrt(x))^2 + 4) = sqrt(x + 4). However, the
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly substitutes u(x) into g(t) as sqrt(x + 4) instead of the correct sqrt(x + 4) derived from sqrt((sqrt(x))^2 + 4). Wait, sqrt
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.