Integrals giving inverse trig functions
Problem 4.634 · medium
Evaluate \( \displaystyle \int \frac{4}{\sqrt{- x^{2} - 2 x + 8}}\, dx \).
- \[ - x^{2} - 2 x + 8 = 9 - \left(x + 1\right)^{2} \]Complete the square.✓ Proved
- With u = x + 1 and a = 3, this is c·∫ du/√(a² − u²)
- \[ \frac{d}{d x} 4 \operatorname{asin}{\left(\frac{x}{3} + \frac{1}{3} \right)} = \frac{4}{\sqrt{- x^{2} - 2 x + 8}} \]An antiderivative is 4*asin(x/3 + 1/3); differentiate to confirm.✓ Proved
Answer \( 4 \operatorname{asin}{\left(\frac{x}{3} + \frac{1}{3} \right)} + C \)
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: fail (misleading) — The sentence claims the integral becomes "c·∫ du/√(a²−u²)" but does not specify the constant factor. In fact, after substituting u=x+1 the integrand is 4/√(9−u²), so the correct statement should be "4∫ du/√(9−u²)". The omission could mislead a student about the role of the constant factor.qwen3.6:27b-mlx: pass — The solution correctly completes the square and identifies the standard integral form. The final verification by differentiation confirms the result.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-07 — The sentence claims the integral becomes "c·∫ du/√(a²−u²)" but does not specify the constant factor. In fact, after substituting u=x+1 the integrand is 4/√(9−u²), so the correct statement should be "4∫ du/√(9−u²)". The omission could mislead a student about the role of the constant factor.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly completes the square and identifies the standard integral form. The final verification by differentiation confirms the result.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly completes the square, identifies the standard integral form, and verifies the result by differentiation. The logic is sound and complete.
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 2026-10-07 with SymPy 1.14.0.