Integral of \( \displaystyle x e^{x^{2}} \)
Problem 4.225 · easy
Find \( \displaystyle \int x e^{x^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int x e^{x^{2}}\, dx \]integral algebraStart with the integral of the function. Rewrite the integrand to expose the derivative of the exponent.✓ Proved
- \[ = \frac{\int 2 x e^{x^{2}}\, dx}{2} \]linearity substitutionPull out the constant factor. Recognize the inner function and its derivative.✓ Proved
- \[ = \frac{e^{x^{2}}}{2} \]antiderivativeIntegrate the remaining expression using the substitution rule.✓ Proved
Answer \( \frac{e^{x^{2}}}{2} + C \)
✓ 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.
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 4 is labeled "substitution" but only reorders the integrand; no change of variables or application of the substitution rule occurs. It should be labeled "rewrite" (or "algebra") instead.qwen3.6:27b-mlx: fail (style) — Step 4 is labeled 'substitution' but performs no algebraic change (it merely reorders terms), violating the 'one thing per step' rule. The actual substitution occurs in Step 5, which is labeled 'antiderivative'.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-03 — Step 4 is labeled 'substitution' but performs no algebraic change (it merely reorders terms), violating the 'one thing per step' rule. The actual substitution occurs in Step 5, which is labeled 'antiderivative'.gpt-oss:20b: fail (style) 2026-10-03 — Step 4 is labeled "substitution" but only reorders the integrand; no change of variables or application of the substitution rule occurs. It should be labeled "rewrite" (or "algebra") instead.qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (style) 2026-10-03 — Step 3 incorrectly labels the extraction of a constant as "linearity"; the proper label is "constant‑multiple". Step 4 also mislabels a simple reordering of factors as "substitution"; no substitution has been performed, so the correct label would be "rewrite" or "algebra".
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 2026-10-03 with SymPy 1.14.0.