Improper integrals
Problem 4.31 · medium
Evaluate \( \displaystyle \int_{0}^{4} \frac{1}{\sqrt[3]{x}} \, dx \), or show that it diverges.
- The integral is improper at x = 0; replace that bound by T and take a limit.
- \[ \frac{d}{d x} \frac{3 x^{\frac{2}{3}}}{2} = \frac{1}{\sqrt[3]{x}} \]An antiderivative, checked by differentiating.✓ Proved
- \[ - \frac{3 T^{\frac{2}{3}}}{2} + 3 \cdot 2^{1 \cdot \frac{1}{3}} = - \frac{3 T^{\frac{2}{3}}}{2} + 3 \sqrt[3]{2} \]The integral with T in place of the bad bound.
- As T → 0, this approaches 3*2**(1/3).
- \[ \lim_{T \to 0^+}\left(- \frac{3 T^{\frac{2}{3}}}{2} + 3 \sqrt[3]{2}\right) = 3 \sqrt[3]{2} \]The limit.✓ Proved
Answer \( 3 \sqrt[3]{2} \)
Lines: 2 proved, 3 not checked. 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 4 | Not checked | — | 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: fail (error) — The antiderivative is correct, but the evaluation at the upper limit is wrong: 4^{2/3}=2^{4/3}=2·2^{1/3}, not 2^{1/3}. Thus the final value 3·2^{1/3} is incorrect.qwen3.6:27b-mlx: pass — The solution correctly identifies the improper nature of the integral, computes the antiderivative, evaluates the definite integral with the variable bound, and correctly calculates the limit as the bound approaches 0.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the improper nature of the integral, computes the antiderivative, evaluates the definite integral with the variable bound, and correctly calculates the limit as the bound approaches 0.gpt-oss:20b: fail (error) 2026-09-28 — The antiderivative is correct, but the evaluation at the upper limit is wrong: 4^{2/3}=2^{4/3}=2·2^{1/3}, not 2^{1/3}. Thus the final value 3·2^{1/3} is incorrect.qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the improper integral, finds the antiderivative, evaluates the definite integral with a limit, and computes the limit 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.