Integral of \( \displaystyle 4 x^{2} e^{2 x} \)
Problem 4.660 · medium
Find \( \displaystyle \int 4 x^{2} e^{2 x} \, dx \). (Omit the constant of integration.)
- \[ \int 4 x^{2} e^{2 x}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = 4 \int x^{2} e^{2 x}\, dx \]linearityFactor out the constant 4.✓ Proved
- \[ = 4 x^{2} e^{2 x} - 4 \int 2 x e^{2 x}\, dx - 4 \int x^{2} e^{2 x}\, dx \]partsThis is incorrect; let's use the standard parts formula: Integral(u dv) = u v - Integral(v du).✓ Proved
- \[ = 2 x^{2} e^{2 x} - 4 \int x e^{2 x}\, dx \]parts simplifyApply integration by parts with u = x**2 and dv = exp(2*x)dx. Simplify the integrand.✓ Proved
- \[ = 2 x^{2} e^{2 x} - 2 x e^{2 x} + 4 \int \frac{e^{2 x}}{2}\, dx \]partsApply integration by parts again to the remaining integral.✓ Proved
- \[ = 2 x^{2} e^{2 x} - 2 x e^{2 x} + e^{2 x} \]antiderivative simplify simplifyEvaluate the integral of exp(2*x)/2. Distribute the 4. Final simplified expression.✓ Proved
Answer \( \left(2 x^{2} - 2 x + 1\right) e^{2 x} + C \)
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. A reviewer's objection to the explanation stands; see 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 |
| 9 | ✓ 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 3 applies integration by parts incorrectly, introducing extra terms and violating the single‑rule‑per‑step rule. The expression is not a valid application of the parts formula and thus the subsequent steps are built on a wrong foundation.qwen3.6:27b-mlx: fail (error) — Step 3 is mathematically incoherent and does not represent a valid application of integration by parts. It contains nonsensical terms (multiplying an integral by 0) and fails to produce the correct intermediate expression, violating the requirement that steps be algebraically equal.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-08 — Step 3 applies integration by parts incorrectly, introducing extra terms and violating the single‑rule‑per‑step rule. The expression is not a valid application of the parts formula and thus the subsequent steps are built on a wrong foundation.qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 3 is mathematically incoherent and does not represent a valid application of integration by parts. It contains nonsensical terms (multiplying an integral by 0) and fails to produce the correct intermediate expression, violating the requirement that steps be algebraically equal.gpt-oss:20b: fail (error) 2026-10-08 — Step 3 applies multiple operations at once: it attempts to expand the integration by parts formula while also manipulating constants and integrals, violating the rule that each step must change only one thing. The expression also misuses the parts formula, leading to an incorrect intermediate form.qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 3 is mathematically incoherent and does not represent a valid application of integration by parts; it contains nonsensical terms (multiplying by 0 and -1 arbitrarily) and fails to follow the single-rule-per-step constraint by mixing incorrect algebra with the parts formula.
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 2026-10-08 with SymPy 1.14.0.