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Definite integrals

Problem 4.128 · easy Video

Evaluate \( \displaystyle \int_{-1}^{0} x \left(x^{2} + 1\right)^{2} \, dx \).

By the Fundamental Theorem of Calculus, Part 2, the integral is F(b) − F(a) for any antiderivative F.
Reviewed
\[ \frac{d}{d x} \left(\frac{x^{6}}{6} + \frac{x^{4}}{2} + \frac{x^{2}}{2}\right) = x \left(x^{2} + 1\right)^{2} \]
F(x) = x**6/6 + x**4/2 + x**2/2 is an antiderivative: its derivative is the integrand.✓ Proved
\[ - \left. \frac{x^{6}}{6} + \frac{x^{4}}{2} + \frac{x^{2}}{2} \right|_{\substack{ x=-1 }} + \left. \frac{x^{6}}{6} + \frac{x^{4}}{2} + \frac{x^{2}}{2} \right|_{\substack{ x=0 }} = - \frac{7}{6} \]
F(b) − F(a).✓ Proved

Answer \( - \frac{7}{6} \)

The graph of f(x) = x·(x² + 1)² from x = -1 to x = 0. Rectangles under the curve get thinner until they fill the area, -7/6. Area below the x-axis counts as negative.

Lines: 2 proved, 1 reviewed. 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	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 identifies an antiderivative and applies the Fundamental Theorem of Calculus to obtain the correct result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies an antiderivative and applies the Fundamental Theorem of Calculus to obtain the correct result.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies an antiderivative and applies the Fundamental Theorem of Calculus. The algebraic verification confirms the antiderivative is correct, and the evaluation yields the stated answer.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The proposed antiderivative F(x) = x^6/6 + x^4/2 + x^2/2 is incorrect; its derivative is x^5 + 2x^3 + x, which does not match the integrand x(x^2 + 1)^2 = x^5 + 2x^3 + x. Wait, actually x(x^2+1)^2 = x(x^4+2x^2+1) = x^5+2x^3+x. The derivative of x^6/6 is x^5, of x^4/2 is 2x^3, of x^2/2 is x. So the derivative IS the integrand. The algebra is correct. Let me re-read the prompt constraints. 'Each line of the solution is either an EQUATION... or a SENTENCE'. Line 2 is marked [equation, unchecked] but the text says 'Derivative(...) ... -- F(x) ... is an antiderivative'. The prompt says 'do not re-check the algebra of a line marked proved'. Line 2 is marked 'unchecked' in the bracket but the description says 'computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. This is contradictory. Let's look at the structure. Line 2 is labeled [equation, unchecked]. The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. It seems Line 2 is treated as an equation line where the CAS checked the derivative. If the CAS checked it, and it is correct, then the step is valid. Line 3 computes F(0) - F(-1). F(0) = 0. F(-1) = 1/6 + 1/2 + 1/2 = 1/6 + 1 = 7/6. F(0) - F(-1) = -7/6. The calculation is correct. The setup is correct. Why would it fail? Let me check the prompt again. 'Derivative(x**6/6 + x**4/2 + x**2/2, x), x*(x**2 + 1)**2'. This asserts the derivative is the integrand. This is true. 'F(b) - F(a)'. This is true. The result is -7/6. This is true. Is there a subtle error? The problem asks to evaluate the integral. The solution provides the correct answer using a valid method. The only potential issue is the labeling. Line 2 is labeled [equation, unchecked]. The prompt says 'do not re-check the algebra of a line marked proved'. It doesn't say 'unchecked' lines are exempt from checking. However, usually 'unchecked' in these prompts implies the CAS did NOT verify it, or it's a raw output. But the text says 'its result is shown'. Let's assume the CAS verified it. If the CAS verified it, it's correct. If the CAS did NOT verify it, I must check it. I checked it, and it is correct. So the math is sound. Is there a style issue? 'By the Fundamental Theorem of Calculus, Part 2...' is a standard sentence. The solution seems perfectly fine. Wait, look at Line 2 again. `Derivative(x**6/6 + x**4/2 + x**2/2, x), x*(x**2 + 1)**2`. This is a tuple or a comparison? It looks like a CAS output format. The prompt says 'do not re-check the algebra of a line marked proved'. Line 2 is marked 'unchecked'. This likely means I SHOULD check it. I did, and it is correct. Line 3 is marked 'unchecked'. I should check it. ` -Subs(...) + Subs(...)`. F(0) - F(-1). F(0)=0. F(-1) = 1/6 + 1/2 + 1/2 = 7/6. Result -7/6. Correct. So why would it fail? Maybe the prompt implies that 'unchecked' means the CAS didn't check it, so I have to. And I did, and it's right. So it should pass. Let me look for any other errors. Did I expand x(x^2+1)^2 correctly? x(x^4+2x^2+1) = x^5+2x^3+x. Derivative of x^6/6 is x^5. Derivative of x^4/2 is 2x^3. Derivative of x^2/2 is x. Sum is x^5+2x^3+x. Yes. Integration limits -1 to 0. F(0)-F(-1). Correct. Answer -7/6. Correct. I will mark it as pass.
gpt-oss:20b: pass 2026-09-28

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 2026-10-03 with SymPy 1.14.0.