Integrals giving inverse trig functions
Problem 4.341 · easy
Evaluate \( \displaystyle \int_{0}^{4} \frac{2}{x^{2} + 16}\, dx \).
- This is c·∫ dx/(x² + a²) with a = 4.Reviewed
- \[ \frac{d}{d x} \frac{\operatorname{atan}{\left(\frac{x}{4} \right)}}{2} = \frac{2}{x^{2} + 16} \]An antiderivative is atan(x/4)/2; differentiate to confirm.✓ Proved
- \[ \frac{\pi}{8} \]Evaluate between the limits.✓ Proved
Answer \( \frac{\pi}{8} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the standard integral form, provides a verified 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, provides a verified antiderivative, and correctly evaluates the definite integral.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the standard integral form, provides the correct antiderivative, and evaluates the definite integral accurately.gpt-oss:20b: pass 2026-10-04
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.