Integrals giving inverse trig functions
Problem 4.335 · easy
Evaluate \( \displaystyle \int_{0}^{\sqrt{3}} \frac{1}{\sqrt{4 - x^{2}}}\, dx \).
- This is c·∫ dx/√(a² − x²) with a = 2.
- \[ \frac{d}{d x} \operatorname{asin}{\left(\frac{x}{2} \right)} = \frac{1}{\sqrt{4 - x^{2}}} \]An antiderivative is asin(x/2); differentiate to confirm.✓ Proved
- \[ \frac{\pi}{3} \]Evaluate between the limits.✓ Proved
Answer \( \frac{\pi}{3} \)
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 (misleading) — The solution skips the essential step of evaluating the antiderivative at the limits; it merely states the result without showing that ∫₀^{√3}1/√(4−x²)dx = π/3. This omission could mislead a student into thinking the answer is automatically verified.qwen3.6:27b-mlx: pass — The solution correctly identifies the standard integral form, verifies the antiderivative, and correctly evaluates the definite integral.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the standard integral form, verifies the antiderivative, and correctly evaluates the definite integral.gpt-oss:20b: fail (misleading) 2026-10-04 — The solution skips the essential step of evaluating the antiderivative at the limits; it merely states the result without showing that ∫₀^{√3}1/√(4−x²)dx = π/3. This omission could mislead a student into thinking the answer is automatically verified.qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the standard integral form, verifies the antiderivative, and correctly evaluates the definite integral.gpt-oss:20b: pass 2026-10-04 — The final sentence is incomplete; it should explicitly evaluate the antiderivative at the bounds to show the result π/3.
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.