Integrals giving inverse trig functions
Problem 4.332 · easy
Evaluate \( \displaystyle \int \frac{1}{x^{2} + 16}\, dx \).
- This is c·∫ dx/(x² + a²) with a = 4.Reviewed
- \[ \frac{d}{d x} \frac{\operatorname{atan}{\left(\frac{x}{4} \right)}}{4} = \frac{1}{x^{2} + 16} \]An antiderivative is atan(x/4)/4; differentiate to confirm.✓ Proved
Answer \( \frac{\operatorname{atan}{\left(\frac{x}{4} \right)}}{4} + C \)
✓ Nihil obstat Lines: 1 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | difference quotient of the answer at x = -4.0 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the standard integral form and verifies the result via differentiation. The setup and conclusion are sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the standard integral form and verifies the result via differentiation. The setup and conclusion are sound.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the standard integral form and verifies the result via differentiation. The setup and conclusion are sound.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.