Definite integrals
Problem 4.201 · medium
Evaluate \( \displaystyle \int_{1}^{e} 4 \sqrt{x} + 2 \, dx \).
- By the Fundamental Theorem of Calculus, Part 2, the integral is F(b) − F(a) for any antiderivative F.
- \[ \frac{d}{d x} \left(\frac{8 x^{\frac{3}{2}}}{3} + 2 x\right) = 4 \sqrt{x} + 2 \]F(x) = 8*x**(3/2)/3 + 2*x is an antiderivative: its derivative is the integrand.✓ Proved
- \[ - \left. 2 x + \frac{8 x^{\frac{3}{2}}}{3} \right|_{\substack{ x=1 }} + \left. \frac{8 x^{\frac{3}{2}}}{3} + 2 x \right|_{\substack{ x=e }} = - \frac{14}{3} + 2 e + \frac{8 e^{\frac{3}{2}}}{3} \]F(b) − F(a).✓ Proved
Answer \( - \frac{14}{3} + 2 e + \frac{8 e^{\frac{3}{2}}}{3} \)
Lines: 2 proved, 1 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.
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 | ✓ 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 (mpmath) gives the same value |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly evaluates the antiderivative at the upper limit x=e, substituting e^(3/2) instead of e^(3/2) derived from e^(3/2). Wait, le
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-09-29 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly evaluates the antiderivative at the upper limit x=e, substituting e^(3/2) instead of e^(3/2) derived from e^(3/2). Wait, legpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: inconclusive 2026-09-29 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly evaluates the antiderivative at the upper bound e, substituting exp(3/2) for e^(3/2). Since e^(3/2) is not equal to exp(3/2gpt-oss:20b: pass 2026-09-29
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/definite_integral, checked 2026-09-29 with SymPy 1.14.0.