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Integrals giving inverse trig functions

Problem 4.333 · medium

Evaluate \( \displaystyle \int_{0}^{2} \frac{4}{\sqrt{- x^{2} + 2 x + 3}}\, dx \).

\[ - x^{2} + 2 x + 3 = 4 - \left(x - 1\right)^{2} \]
Complete the square.✓ Proved
With u = x − 1 and a = 2, this is c·∫ du/√(a² − u²)
Reviewed
\[ \frac{d}{d x} 4 \operatorname{asin}{\left(\frac{x}{2} - \frac{1}{2} \right)} = \frac{4}{\sqrt{- x^{2} + 2 x + 3}} \]
An antiderivative is 4*asin(x/2 - 1/2); differentiate to confirm.✓ Proved
\[ \frac{4 \pi}{3} \]
Evaluate between the limits.✓ Proved

Answer \( \frac{4 \pi}{3} \)

Lines: 3 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
4	✓ 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	numerical quadrature

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative via completing the square and substitution, and the final evaluation is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the antiderivative via completing the square and substitution, and the final evaluation is correct.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly completes the square, identifies the standard arcsine integral form, and evaluates the definite integral accurately.
gpt-oss:20b: fail (misleading) 2026-10-04 — The sentence claims the integral becomes c·∫du/√(a²−u²) with a=2, but it omits the crucial constant factor 4 that comes from the original integrand. It also fails to mention the transformed limits, so a student could incorrectly assume the integral is simply ∫du/√(4−u²).

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 2026-10-04 with SymPy 1.14.0.