Integral of \( \displaystyle - 4 x + 3 \left(x - 1\right)^{2} + 5 \)
Problem 4.916 · medium
Find \( \displaystyle \int - 4 x + 3 \left(x - 1\right)^{2} + 5 \, dx \). (Omit the constant of integration.)
- \[ \int \left(- 4 x + 3 \left(x - 1\right)^{2} + 5\right)\, dx \]integralStart with the integral of the given expression.✓ Proved
- \[ = \int \left(3 x^{2} - 10 x + 8\right)\, dx \]algebra algebra simplifyExpand the squared term. Distribute the 3. Combine like terms.✓ Proved
- \[ = \int 8\, dx - \int 10 x\, dx + \int 3 x^{2}\, dx \]linearitySplit the integral into three parts.✓ Proved
- \[ = x^{3} - 5 x^{2} + 8 x \]antiderivativeIntegrate each term using the power rule.✓ Proved
Answer \( x \left(x^{2} - 5 x + 8\right) + C \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly expands, simplifies, and integrates the polynomial term by term. Each step applies a single rule from the allowed vocabulary, and the labels are accurate.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly expands, simplifies, and integrates the polynomial term by term. Each step applies a single rule from the allowed vocabulary, and the labels are accurate.qwen3.6:27b-mlx: pass 2026-10-12gpt-oss:20b: pass 2026-10-12
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
gemma4:26b, checked 2026-10-12 with SymPy 1.14.0.