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Integrals giving inverse trig functions

Problem 4.810 · medium

Evaluate \( \displaystyle \int \frac{1}{\sqrt{- x^{2} - 2 x + 8}}\, dx \).

\[ - x^{2} - 2 x + 8 = 9 - \left(x + 1\right)^{2} \]
Complete the square.✓ Proved
With u = x + 1 and a = 3, this is c·∫ du/√(a² − u²)
\[ \frac{d}{d x} \operatorname{asin}{\left(\frac{x}{3} + \frac{1}{3} \right)} = \frac{1}{\sqrt{- x^{2} - 2 x + 8}} \]
An antiderivative is asin(x/3 + 1/3); differentiate to confirm.✓ Proved

Answer \( \operatorname{asin}{\left(\frac{x}{3} + \frac{1}{3} \right)} + C \)

Lines: 2 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	a sentence; read, not computed
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	difference quotient of the answer at x = -3.9

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (misleading) — The solution omits the constant factor c=1 in the transition from step 1 to step 2, implying the integral is simply ∫ du/√(a² − u²) without explicitly stating that the coefficient is 1. While the final answer is correct, the notation 'c·∫' suggests a generic constant that needs to be determined, but the text fails to explicitly state that c=1, which could confuse a student about where the coefficient comes from or if it was dropped.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (misleading) 2026-10-10 — The solution omits the constant factor c=1 in the transition from step 1 to step 2, implying the integral is simply ∫ du/√(a² − u²) without explicitly stating that the coefficient is 1. While the final answer is correct, the notation 'c·∫' suggests a generic constant that needs to be determined, but the text fails to explicitly state that c=1, which could confuse a student about where the coefficient comes from or if it was dropped.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly completes the square, identifies the standard integral form, and verifies the result by differentiation. The constant factor is handled implicitly in the final verification step.

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.