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Definite integrals

Problem 4.93 · easy Video

Evaluate \( \displaystyle \int_{-1}^{1} x \left(x^{2} + 1\right)^{3} \, dx \).

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
\[ - \left. \frac{x^{8}}{8} + \frac{x^{6}}{2} + \frac{3 x^{4}}{4} + \frac{x^{2}}{2} \right|_{\substack{ x=-1 }} + \left. \frac{x^{8}}{8} + \frac{x^{6}}{2} + \frac{3 x^{4}}{4} + \frac{x^{2}}{2} \right|_{\substack{ x=1 }} = 0 \]
F(b) − F(a).✓ Proved

Answer \( 0 \)

The graph of f(x) = x·(x² + 1)³ from x = -1 to x = 1. Rectangles under the curve get thinner until they fill the area, 0. Area below the x-axis counts as negative.

✓ 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. 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	✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the Fundamental Theorem of Calculus. The identified antiderivative is correct, and the evaluation at the bounds yields the correct result of 0.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the Fundamental Theorem of Calculus. The identified antiderivative is correct, and the evaluation at the bounds yields the correct result of 0.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the Fundamental Theorem of Calculus. The antiderivative is verified, and the evaluation at the bounds is correct.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies an antiderivative and applies the Fundamental Theorem of Calculus to obtain the correct result. The algebraic steps are verified, and the logic is sound.
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 2026-09-27 with SymPy 1.14.0.