Definite integrals
Problem 4.88 · easy
Evaluate \( \displaystyle \int_{-1}^{2} x \left(x^{2} + 1\right)^{3} \, dx \).
- By the Fundamental Theorem of Calculus, Part 2, the integral is F(b) − F(a) for any antiderivative F.Reviewed
- \[ \frac{d}{d x} \left(\frac{x^{8}}{8} + \frac{x^{6}}{2} + \frac{3 x^{4}}{4} + \frac{x^{2}}{2}\right) = x \left(x^{2} + 1\right)^{3} \]F(x) = x**8/8 + x**6/2 + 3*x**4/4 + x**2/2 is an antiderivative: its derivative is the integrand.✓ Proved
- \[ - \left. \frac{x^{8}}{8} + \frac{x^{6}}{2} + \frac{3 x^{4}}{4} + \frac{x^{2}}{2} \right|_{\substack{ x=-1 }} + \left. \frac{x^{8}}{8} + \frac{x^{6}}{2} + \frac{3 x^{4}}{4} + \frac{x^{2}}{2} \right|_{\substack{ x=2 }} = \frac{609}{8} \]F(b) − F(a).✓ Proved
Answer \( \frac{609}{8} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 | ✓ 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: pass — The solution correctly identifies an antiderivative and applies the Fundamental Theorem of Calculus. The algebraic verification confirms the antiderivative is correct, and the final evaluation is accurate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies an antiderivative and applies the Fundamental Theorem of Calculus. The algebraic verification confirms the antiderivative is correct, and the final evaluation is accurate.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies an antiderivative by expanding the integrand and applying the Fundamental Theorem of Calculus. The arithmetic for F(2) - F(-1) is correct.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/definite_integral, checked 2026-09-26 with SymPy 1.14.0.