Integral of \( \displaystyle 3 x e^{3 x} \)
Problem 4.1340 · medium Video
Find \( \displaystyle \int 3 x e^{3 x} \, dx \). (Omit the constant of integration.)
- \[ \int 3 x e^{3 x}\, dx \]integral partsStart with the integral of the function. Set up integration by parts with u = x and dv = 3*exp(3*x) dx.✓ Proved
- \[ = x e^{3 x} - \int e^{3 x}\, dx \]partsApply the integration by parts formula: Integral(u dv) = uv - Integral(v du).✓ Proved
- \[ = x e^{3 x} - \frac{e^{3 x}}{3} \]antiderivativeEvaluate the remaining integral.✓ Proved
- \[ = \left(x - \frac{1}{3}\right) e^{3 x} \]simplifyFactor out the common exponential term.✓ Proved
Answer \( \left(x - \frac{1}{3}\right) e^{3 x} + C \)
✓ 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 |
| 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: fail (style) — Step 2 incorrectly labels the rewrite of 3*x*exp(3*x) as "parts"; it should be a "rewrite" step, and the note about setting up integration by parts is misleading because the actual integration by parts occurs in step 3.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-26gpt-oss:20b: fail (style) 2026-10-26 — Step 2 incorrectly labels the rewrite of 3*x*exp(3*x) as "parts"; it should be a "rewrite" step, and the note about setting up integration by parts is misleading because the actual integration by parts occurs in step 3.qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly applies integration by parts and simplifies the result. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-10-25qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly applies integration by parts and standard antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-25
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 2026-10-26 with SymPy 1.14.0.