Integral of \( \displaystyle \left(3 x + 1\right) e^{3 x + 1} \)
Problem 4.796 · medium
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
- \[ = \int 3 x e^{3 x + 1}\, dx + \int e^{3 x + 1}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = \frac{e^{3 x + 1}}{3} + \int 3 x e^{3 x + 1}\, dx \]antiderivativeEvaluate the second integral.✓ Proved
- \[ = x e^{3 x + 1} + \frac{e^{3 x + 1}}{3} - \int e^{3 x + 1}\, dx \]parts simplifyApply integration by parts to the first integral with u = 3*x and dv = exp(3*x + 1)dx. Simplify the terms from the integration by parts step.✓ Proved
- \[ = x e^{3 x + 1} \]antiderivative simplifyEvaluate the remaining integral. Combine the terms to find the final antiderivative.✓ Proved
Answer \( x e^{3 x + 1} + C \)
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.
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: fail (error) — Step 5 incorrectly applies integration by parts: it adds an extra +exp(3*x+1)/3 term that should not appear. The correct result of the parts step is 3*x*(exp(3*x+1)/3) - Integral(exp(3*x+1),x), without the additional term.qwen3.6:27b-mlx: pass — The solution correctly applies linearity, integration by parts, and simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-10 — Step 5 incorrectly applies integration by parts: it adds an extra +exp(3*x+1)/3 term that should not appear. The correct result of the parts step is 3*x*(exp(3*x+1)/3) - Integral(exp(3*x+1),x), without the additional term.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies linearity, integration by parts, and simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies integration by parts and linearity. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
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 2026-10-10 with SymPy 1.14.0.