Integrals giving inverse trig functions
Problem 4.640 · easy
Evaluate \( \displaystyle \int_{0}^{2} \frac{1}{x^{2} + 4}\, dx \).
- This is c·∫ dx/(x² + a²) with a = 2.
- \[ \frac{d}{d x} \frac{\operatorname{atan}{\left(\frac{x}{2} \right)}}{2} = \frac{1}{x^{2} + 4} \]An antiderivative is atan(x/2)/2; differentiate to confirm.✓ Proved
- \[ \frac{\pi}{8} \]Evaluate between the limits.✓ Proved
Answer \( \frac{\pi}{8} \)
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 omits the crucial step of evaluating the antiderivative at the limits 0 and 2. It merely states the final value without showing that atan(2/2)/2 – atan(0/2)/2 equals π/8, so the reasoning is incomplete and potentially misleading.qwen3.6:27b-mlx: pass — The solution correctly identifies the standard integral form, verifies the antiderivative, and computes the definite integral accurately.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — The solution omits the crucial step of evaluating the antiderivative at the limits 0 and 2. It merely states the final value without showing that atan(2/2)/2 – atan(0/2)/2 equals π/8, so 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 computes the definite integral accurately.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the standard integral form and applies the appropriate antiderivative. The evaluation of the definite integral is correct.
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.