Integrals giving inverse trig functions
Problem 4.506 · easy
Evaluate \( \displaystyle \int_{-5}^{5} \frac{2}{x^{2} + 25}\, dx \).
- This is c·∫ dx/(x² + a²) with a = 5.
- \[ \frac{d}{d x} \frac{2 \operatorname{atan}{\left(\frac{x}{5} \right)}}{5} = \frac{2}{x^{2} + 25} \]An antiderivative is 2*atan(x/5)/5; differentiate to confirm.✓ Proved
- \[ \frac{\pi}{5} \]Evaluate between the limits.✓ Proved
Answer \( \frac{\pi}{5} \)
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 omits the evaluation step: it states "pi/5 = pi/5" without showing how the antiderivative is applied to the limits. Additionally, the first sentence does not specify the constant factor c=2, so the setup is incomplete.qwen3.6:27b-mlx: pass — The solution correctly identifies the standard integral form, provides the correct antiderivative, and evaluates the definite integral accurately.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-06 — The solution omits the evaluation step: it states "pi/5 = pi/5" without showing how the antiderivative is applied to the limits. Additionally, the first sentence does not specify the constant factor c=2, so the setup is incomplete.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the standard integral form, provides the correct antiderivative, and evaluates the definite integral accurately.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the standard integral form, verifies the antiderivative, and correctly evaluates the definite integral using the limits of integration.
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.