Integrals giving inverse trig functions
Problem 4.507 · medium
Evaluate \( \displaystyle \int \frac{1}{\sqrt{- x^{2} - 2 x + 3}}\, dx \).
- \[ - x^{2} - 2 x + 3 = 4 - \left(x + 1\right)^{2} \]Complete the square.✓ Proved
- With u = x + 1 and a = 2, this is c·∫ du/√(a² − u²)
- \[ \frac{d}{d x} \operatorname{asin}{\left(\frac{x}{2} + \frac{1}{2} \right)} = \frac{1}{\sqrt{- x^{2} - 2 x + 3}} \]An antiderivative is asin(x/2 + 1/2); differentiate to confirm.✓ Proved
Answer \( \operatorname{asin}{\left(\frac{x}{2} + \frac{1}{2} \right)} + C \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 = -2.9 |
Reviewers
gpt-oss:20b: fail (style) — [domain objection, downgraded to style] The sentence claims the integral becomes "c·∫ du/√(a²−u²)" with an undefined constant c. In fact, after the substitution u=x+1, the integral is exactly ∫du/√(4−u²) with no extra factor. This misstates the standard form and could mislead a student about the role of constants in the substitution.qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-06 — [domain objection, downgraded to style] The sentence claims the integral becomes "c·∫ du/√(a²−u²)" with an undefined constant c. In fact, after the substitution u=x+1, the integral is exactly ∫du/√(4−u²) with no extra factor. This misstates the standard form and could mislead a student about the role of constants in the substitution.qwen3.6:27b-mlx: pass 2026-10-06gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — 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 2026-10-06 with SymPy 1.14.0.