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Improper integrals

Problem 4.32 · easy

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

The integral is improper at infinity; replace that bound by T and take a limit.
Reviewed
\[ \frac{d}{d x} 2 \sqrt{x} = \frac{1}{\sqrt{x}} \]
An antiderivative, checked by differentiating.✓ Proved
\[ 2 \sqrt{T} - 2 \]
The integral with T in place of the bad bound.Reviewed
As T → \infty, this grows without bound, so the integral diverges.
Reviewed

Answer \( \text{diverges} \)

✓ Nihil obstat Lines: 1 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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	numerical integrals to 10², 10⁴, 10⁶ keep growing

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the improper integral, finds the antiderivative, evaluates the definite integral with the limit variable, and correctly concludes divergence based on the limit behavior.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the improper integral, finds the antiderivative, evaluates the definite integral with the limit variable, and correctly concludes divergence based on the limit behavior.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the improper nature of the integral, finds the correct antiderivative, evaluates the definite integral with the limit variable, and correctly concludes divergence based on the limit behavior.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the improper nature of the integral, finds the correct antiderivative, and properly evaluates the limit to conclude divergence.
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.