Integral of \( \displaystyle 6 x + \left(3 x + 1\right)^{3} + 2 \)
Problem 4.1214 · medium Video
Find \( \displaystyle \int 6 x + \left(3 x + 1\right)^{3} + 2 \, dx \). (Omit the constant of integration.)
- \[ \int \left(6 x + \left(3 x + 1\right)^{3} + 2\right)\, dx \]integralStart with the integral of the given expression.✓ Proved
- \[ = \int \left(27 x^{3} + 27 x^{2} + 15 x + 3\right)\, dx \]algebra algebra simplifyExpand the cubic term (3x + 1)^3. Simplify the expanded terms. Combine like terms.✓ Proved
- \[ = \int 3\, dx + \int 15 x\, dx + \int 27 x^{2}\, dx + \int 27 x^{3}\, dx \]linearitySplit the integral into individual terms.✓ Proved
- \[ = \frac{27 x^{4}}{4} + 9 x^{3} + \frac{15 x^{2}}{2} + 3 x \]antiderivativeIntegrate each term using the power rule.✓ Proved
Answer \( \frac{3 x \left(9 x^{3} + 12 x^{2} + 10 x + 4\right)}{4} + C \)
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. 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 | ✓ 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 the polynomial, combines like terms, applies linearity, and integrates term-by-term. The final answer matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly expands the polynomial, combines like terms, applies linearity, and integrates term-by-term. The final answer matches the stated answer.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — 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 appropriate.gpt-oss:20b: fail (error) 2026-10-16 — Step 3 incorrectly expands (3*x + 1)**3, replacing the constant term 1 with 3, leading to an incorrect integrand.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly expands the cubic term, simplifies the polynomial, applies linearity, and integrates term-by-term. The final answer matches the stated answer.gpt-oss:20b: fail (error) 2026-10-15 — Step 3 incorrectly replaces the constant term 1 from the expansion of (3x+1)^3 with 3, leading to an incorrect integrand. The correct expansion is 27*x**3 + 27*x**2 + 9*x + 1, so the subsequent steps are based on a wrong expression.
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 2026-10-18 with SymPy 1.14.0.