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Integrals giving inverse trig functions

Problem 4.510 · easy

Evaluate \( \displaystyle \int_{0}^{4 \sqrt{3}} \frac{1}{x^{2} + 16}\, dx \).

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
\[ \frac{\pi}{12} \]
Evaluate between the limits.✓ Proved

Answer \( \frac{\pi}{12} \)

Lines: 2 proved, 1 reviewed. 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	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: pass
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: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the standard integral form, verifies the antiderivative, and computes the definite integral accurately.
gpt-oss:20b: fail (misleading) 2026-10-06 — The solution never evaluates the antiderivative at the limits 0 and 4√3; it merely states an equality that does not follow from the preceding steps. The conclusion that the integral equals π/12 is unsupported and would mislead a student.
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the standard integral form and applies the appropriate antiderivative. The evaluation of the limits 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 2026-10-06 with SymPy 1.14.0.