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Improper integrals

Problem 4.35 · medium

Evaluate \( \displaystyle \int_{0}^{4} \frac{1}{x^{\frac{2}{3}}} \, dx \), or show that it diverges.

The integral is improper at x = 0; replace that bound by T and take a limit.
Reviewed
\[ \frac{d}{d x} 3 \sqrt[3]{x} = \frac{1}{x^{\frac{2}{3}}} \]
An antiderivative, checked by differentiating.✓ Proved
\[ - 3 T^{1 \cdot \frac{1}{3}} + 3 \cdot 2^{\frac{2}{3}} = - 3 \sqrt[3]{T} + 3 \cdot 2^{\frac{2}{3}} \]
The integral with T in place of the bad bound.Reviewed
As T → 0, this approaches 3*2**(2/3).
Reviewed
\[ \lim_{T \to 0^+}\left(- 3 \sqrt[3]{T} + 3 \cdot 2^{\frac{2}{3}}\right) = 3 \cdot 2^{\frac{2}{3}} \]
The limit.✓ Proved

Answer \( 3 \cdot 2^{\frac{2}{3}} \)

✓ 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 singularity, computes the antiderivative, sets up the limit properly, and evaluates it correctly.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the singularity, computes the antiderivative, sets up the limit properly, and evaluates it correctly.
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 parameter, and computes the limit correctly.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the singularity, finds the antiderivative, sets up the limit, and evaluates it correctly.
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.