Integral of \( \displaystyle \left(3 x - 1\right)^{2} e^{3 x - 1} \)
Problem 4.1442 · medium Video
Find \( \displaystyle \int \left(3 x - 1\right)^{2} e^{3 x - 1} \, dx \). (Omit the constant of integration.)
- \[ \int \left(3 x - 1\right)^{2} e^{3 x - 1}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(9 x^{2} - 6 x + 1\right) e^{3 x - 1}\, dx \]algebraExpand the squared term.✓ Proved
- \[ = - 6 \int x e^{3 x - 1}\, dx + 9 \int x^{2} e^{3 x - 1}\, dx + \int e^{3 x - 1}\, dx \]linearityDistribute the integral over the sum.✓ Proved
- \[ = 3 x^{2} e^{3 x - 1} - 9 \int \frac{2 x e^{3 x - 1}}{3}\, dx - 6 \int x e^{3 x - 1}\, dx + \int e^{3 x - 1}\, dx \]partsApply integration by parts to the first term.✓ Proved
- \[ = 3 x^{2} e^{3 x - 1} - 12 \int x e^{3 x - 1}\, dx + \int e^{3 x - 1}\, dx \]algebra algebraDistribute the 9 and simplify. Combine the integral terms.✓ Proved
- \[ = 3 x^{2} e^{3 x - 1} - 4 x e^{3 x - 1} + 12 \int \frac{e^{3 x - 1}}{3}\, dx + \int e^{3 x - 1}\, dx \]partsApply integration by parts to the second term.✓ Proved
- \[ = 3 x^{2} e^{3 x - 1} - 4 x e^{3 x - 1} + 5 \int e^{3 x - 1}\, dx \]algebra algebraDistribute the -12 and simplify. Combine the remaining integral terms.✓ Proved
- \[ = 3 x^{2} e^{3 x - 1} - 4 x e^{3 x - 1} + \frac{5 e^{3 x - 1}}{3} \]antiderivativeEvaluate the final integral.✓ Proved
- \[ = \left(3 x^{2} - 4 x + \frac{5}{3}\right) e^{3 x - 1} \]simplifyFactor out the exponential term.✓ Proved
Answer \( \frac{\left(9 x^{2} - 12 x + 5\right) e^{3 x - 1}}{3} + C \)
✓ 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 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ 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
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-06gpt-oss:20b: pass 2026-11-06qwen3.6:27b-mlx: pass 2026-11-05 — The solution correctly applies integration by parts and algebraic simplification at each step. The labels accurately reflect the operations performed, and the final result matches the stated answer.gpt-oss:20b: pass 2026-11-05qwen3.6:27b-mlx: pass 2026-11-05 — The solution correctly applies integration by parts and algebraic simplification at each step. The labels accurately reflect the operations performed, and the final result matches the stated answer.gpt-oss:20b: pass 2026-11-05
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 2026-11-06 with SymPy 1.14.0.