Integrals giving inverse trig functions
Problem 4.637 · easy
Evaluate \( \displaystyle \int_{- \frac{3}{2}}^{\frac{3}{2}} \frac{4}{\sqrt{9 - x^{2}}}\, dx \).
- This is c·∫ dx/√(a² − x²) with a = 3.
- \[ \frac{d}{d x} 4 \operatorname{asin}{\left(\frac{x}{3} \right)} = \frac{4}{\sqrt{9 - x^{2}}} \]An antiderivative is 4*asin(x/3); differentiate to confirm.✓ Proved
- \[ \frac{4 \pi}{3} \]Evaluate between the limits.✓ Proved
Answer \( \frac{4 \pi}{3} \)
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 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | numerical quadrature |
Reviewers
gpt-oss:20b: fail (error) — The solution skips the crucial step of evaluating the antiderivative at the limits. It merely states the final value without showing that arcsin(1/2)=π/6 and arcsin(−1/2)=−π/6, leading to 4π/3. This omission means the reasoning is incomplete and potentially misleading.qwen3.6:27b-mlx: pass — The solution correctly identifies the standard integral form, verifies the antiderivative, and correctly evaluates the definite integral using the Fundamental Theorem of Calculus.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — The solution skips the crucial step of evaluating the antiderivative at the limits. It merely states the final value without showing that arcsin(1/2)=π/6 and arcsin(−1/2)=−π/6, leading to 4π/3. This omission means the reasoning is incomplete and potentially misleading.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the standard integral form, verifies the antiderivative, and correctly evaluates the definite integral using the Fundamental Theorem of Calculus.gpt-oss:20b: fail (error) 2026-10-07 — The definite integral evaluates to 4π, not 4π/3. The conclusion in sentence 3 is incorrect.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the standard integral form, verifies the antiderivative, and applies the Fundamental Theorem of Calculus with correct arithmetic.
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.