Integral of \( \displaystyle 2 x + \left(x + 1\right)^{3} + 2 \)
Problem 4.555 · medium
Find \( \displaystyle \int 2 x + \left(x + 1\right)^{3} + 2 \, dx \). (Omit the constant of integration.)
- \[ \int \left(2 x + \left(x + 1\right)^{3} + 2\right)\, dx \]integralStart with the integral of the given expression.✓ Proved
- \[ = \int \left(x^{3} + 3 x^{2} + 5 x + 3\right)\, dx \]algebra simplifyExpand the binomial (x + 1)**3. Combine like terms.✓ Proved
- \[ = \int 3\, dx + \int 5 x\, dx + \int 3 x^{2}\, dx + \int x^{3}\, dx \]linearitySplit the integral into individual terms.✓ Proved
- \[ = \frac{x^{4}}{4} + x^{3} + \frac{5 x^{2}}{2} + 3 x \]antiderivativeIntegrate each term using the power rule.✓ Proved
Answer \( \frac{x \left(x + 2\right) \left(x^{2} + 2 x + 6\right)}{4} + C \)
✓ 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 |
| 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 applies algebraic expansion, simplification, linearity of integration, and the power rule for antiderivatives. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies algebraic expansion, simplification, linearity of integration, and the power rule for antiderivatives. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies expansion, simplification, linearity, and the power rule for integration in distinct steps. The final result matches the stated answer.
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 2026-10-07 with SymPy 1.14.0.