Integral of \( \displaystyle \left(3 x + 1\right)^{2} e^{\left(3 x + 1\right)^{3}} \)
Problem 4.924 · medium
Find \( \displaystyle \int \left(3 x + 1\right)^{2} e^{\left(3 x + 1\right)^{3}} \, dx \). (Omit the constant of integration.)
- \[ \int \left(3 x + 1\right)^{2} e^{\left(3 x + 1\right)^{3}}\, dx \]integral algebra algebraStart with the integral of the given function. Multiply and divide by 9 to prepare for integration. Rewrite 9 as 3 * 3.✓ Proved
- \[ = \frac{e^{\left(3 x + 1\right)^{3}}}{9} \]antiderivativeThe integrand is the derivative of the expression.✓ Proved
Answer \( \frac{e^{\left(3 x + 1\right)^{3}}}{9} + C \)
✓ 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 |
| 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: fail (style) — Step 3 applies two algebraic manipulations (rewriting 9 as 3*3 and regrouping factors) simultaneously, violating the one-change-per-step rule. Step 4 skips the substitution step and the extraction of the constant factor, jumping directly from the integral to the result.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (style) 2026-10-12 — Step 3 applies two algebraic manipulations (rewriting 9 as 3*3 and regrouping factors) simultaneously, violating the one-change-per-step rule. Step 4 skips the substitution step and the extraction of the constant factor, jumping directly from the integral to the result.qwen3.6:27b-mlx: fail (style) 2026-10-12 — Step 3 is redundant and applies an unnecessary algebraic rewrite that does not advance the solution. Step 4 skips the explicit substitution step required by the 'substitution' rule in the vocabulary, jumping directly to the result.gpt-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.