Integral of \( \displaystyle 2 x e^{4 x^{2}} \)
Problem 4.363 · medium
Find \( \displaystyle \int 2 x e^{4 x^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int 2 x e^{4 x^{2}}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \frac{\int 8 x e^{4 x^{2}}\, dx}{4} \]rewrite algebra substitutionRewrite the integrand to facilitate substitution. This step is a placeholder for the substitution logic. Let u = 4*x**2, then du = 8*x dx.✓ Proved
- \[ = \frac{e^{4 x^{2}}}{4} \]antiderivativeIntegrate the resulting exponential function.✓ Proved
Answer \( \frac{e^{4 x^{2}}}{4} + C \)
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 |
| 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: fail (error) — Step 3 is mathematically nonsensical, adding and subtracting the same term without justification or purpose. Step 4 incorrectly labels the substitution process; the step merely restates the integral from Step 2 without performing the variable change, and the transition to Step 5 skips the actual integration logic.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 3 is mathematically nonsensical, adding and subtracting the same term without justification or purpose. Step 4 incorrectly labels the substitution process; the step merely restates the integral from Step 2 without performing the variable change, and the transition to Step 5 skips the actual integration logic.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 3 is mathematically nonsensical (adding and subtracting the same term) and does not constitute a valid substitution step. Step 4 claims to perform a substitution but leaves the expression unchanged, failing to actually execute the variable change or integrate.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.