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Improper integrals

Problem 4.39 · medium

Evaluate \( \displaystyle \int_{0}^{\infty} \frac{1}{x^{2} + 9} \, dx \), or show that it diverges.

The integral is improper at infinity; replace that bound by T and take a limit.
Reviewed
\[ \frac{d}{d x} \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{3} = \frac{1}{x^{2} + 9} \]
An antiderivative, checked by differentiating.✓ Proved
\[ \frac{\operatorname{atan}{\left(\frac{T}{3} \right)}}{3} \]
The integral with T in place of the bad bound.Reviewed
As T → \infty, this approaches pi/6.
Reviewed
\[ \lim_{T \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{T}{3} \right)}}{3}\right) = \frac{\pi}{6} \]
The limit.✓ Proved

Answer \( \frac{\pi}{6} \)

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
5	✓ 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 over the whole interval agrees

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the improper integral, finds the antiderivative, and evaluates the limit at infinity. The logic is sound and the result is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the improper integral, finds the antiderivative, and evaluates the limit at infinity. The logic is sound and the result is correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the improper integral, finds the correct antiderivative, and evaluates the limit at infinity accurately.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the improper integral, finds the correct antiderivative, and evaluates the limit at infinity accurately.
gpt-oss:20b: pass 2026-09-26

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/improper_integral, checked 2026-10-13 with SymPy 1.14.0.