Integral of \( \displaystyle 2 x e^{- 2 x} \)
Problem 4.1387 · medium Video
Find \( \displaystyle \int 2 x e^{- 2 x} \, dx \). (Omit the constant of integration.)
- \[ \int 2 x e^{- 2 x}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = - x e^{- 2 x} - \int \left(- e^{- 2 x}\right)\, dx \]partsApply integration by parts with u = x and dv = 2*exp(-2*x)dx.✓ Proved
- \[ = - x e^{- 2 x} + \int e^{- 2 x}\, dx \]algebraSimplify the signs.✓ Proved
- \[ = - x e^{- 2 x} - \frac{e^{- 2 x}}{2} \]antiderivativeEvaluate the final integral.✓ Proved
Answer \( \left(- x - \frac{1}{2}\right) e^{- 2 x} + C \)
✓ 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 |
| 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 2026-10-30gpt-oss:20b: pass 2026-10-30qwen3.6:27b-mlx: pass 2026-10-29gpt-oss:20b: pass 2026-10-29qwen3.6:27b-mlx: pass 2026-10-29gpt-oss:20b: pass 2026-10-29
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 2026-10-30 with SymPy 1.14.0.