Integral of \( \displaystyle \left(3 x + 1\right) e^{- 3 x - 1} \)
Problem 4.1311 · medium Video
Find \( \displaystyle \int \left(3 x + 1\right) e^{- 3 x - 1} \, dx \). (Omit the constant of integration.)
- \[ \int \left(3 x + 1\right) e^{- 3 x - 1}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(3 x e^{- 3 x - 1} + e^{- 3 x - 1}\right)\, dx \]linearityDistribute the exponential term.✓ Proved
- \[ = 3 \int x e^{- 3 x - 1}\, dx + \int e^{- 3 x - 1}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = - x e^{- 3 x - 1} - 3 \int \left(- \frac{e^{- 3 x - 1}}{3}\right)\, dx + \int e^{- 3 x - 1}\, dx \]partsApply integration by parts to the first integral term.✓ Proved
- \[ = - x e^{- 3 x - 1} + 2 \int e^{- 3 x - 1}\, dx \]algebra algebraSimplify the integration by parts expression. Combine the integral terms.✓ Proved
- \[ = - x e^{- 3 x - 1} - \frac{2 e^{- 3 x - 1}}{3} \]antiderivative simplifyEvaluate the remaining integral. Simplify the final expression.✓ Proved
Answer \( \left(- x - \frac{2}{3}\right) e^{- 3 x - 1} + C \)
✓ 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. 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 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ 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 linearity, integration by parts, and basic antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly applies linearity, integration by parts, and basic antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly applies linearity, integration by parts, and antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (style) 2026-10-21 — Step 4 applies both the parts rule and an algebraic scaling by 3 in a single step, violating the one‑rule‑per‑step rule. It should be split into two separate steps: first perform integration by parts, then multiply the result by 3.qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly applies linearity and integration by parts. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-21
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 2026-10-22 with SymPy 1.14.0.