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Integrals giving inverse trig functions

Problem 4.807 · easy

Evaluate \( \displaystyle \int \frac{1}{\sqrt{25 - x^{2}}}\, dx \).

This is c·∫ dx/√(a² − x²) with a = 5.
\[ \frac{d}{d x} \operatorname{asin}{\left(\frac{x}{5} \right)} = \frac{1}{\sqrt{25 - x^{2}}} \]
An antiderivative is asin(x/5); differentiate to confirm.✓ Proved

Answer \( \operatorname{asin}{\left(\frac{x}{5} \right)} + C \)

Lines: 1 proved, 1 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.

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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	difference quotient of the answer at x = -4.0

Reviewers

gpt-oss:20b: fail (misleading) — Sentence 1 incorrectly introduces an extraneous constant factor "c·"; the integral is exactly ∫dx/√(25−x²) with a=5, not multiplied by a constant.
qwen3.6:27b-mlx: pass — The solution correctly identifies the standard integral form and verifies the result via differentiation. The logic is sound and complete.

Every verdict on record (4)

gpt-oss:20b: fail (misleading) 2026-10-10 — Sentence 1 incorrectly introduces an extraneous constant factor "c·"; the integral is exactly ∫dx/√(25−x²) with a=5, not multiplied by a constant.
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the standard integral form and verifies the result via differentiation. The logic is sound and complete.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the standard integral form and verifies the result via differentiation. The steps are sound and the conclusion is correct.

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/inverse_trig_integral, checked 2026-10-10 with SymPy 1.14.0.