Definite integrals
Problem 4.128 · easy
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 \( - \frac{7}{6} \)
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.
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 evaluation yields the stated answer.
Every verdict on record (4)
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 2026-09-28qwen3.6:27b-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 2026-09-28 with SymPy 1.14.0.