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Integrals giving inverse trig functions

Problem 4.813 · medium

Evaluate \( \displaystyle \int \frac{3}{\sqrt{- x^{2} + 2 x}}\, dx \).

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

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

Lines: 2 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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 = 0.1

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly completes the square, identifies the standard integral form, and verifies the result by differentiation. The steps are logically sound and mathematically correct.

Every verdict on record (4)

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 steps are logically sound and mathematically correct.
gpt-oss:20b: fail (misleading) 2026-10-10 — The sentence claiming the integral becomes "c·∫ du/√(a²−u²)" omits the crucial factor 3 that comes from the original integrand. This could mislead a student into thinking the constant of integration or the coefficient is irrelevant.
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly completes the square, identifies the standard arcsine integral form, and verifies the result by differentiation.

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.