Integral of \( \displaystyle x e^{x} \)
Problem 4.254 · easy
Find \( \displaystyle \int x e^{x} \, dx \). (Omit the constant of integration.)
- \[ \int x e^{x}\, dx \]integral partsStart with the integral of the integrand. Apply integration by parts where u = x and dv = exp(x)dx.✓ Proved
- \[ = - \int 0\, dx + \int x e^{x}\, dx \]algebra simplifyApply the integration by parts formula. Simplify the remaining integral term.✓ Proved
- \[ = \int x e^{x}\, dx \]simplifyThe integral of zero is zero.✓ Proved
- \[ = x e^{x} - \int e^{x}\, dx \]partsApply integration by parts to the remaining integral.✓ Proved
- \[ = x e^{x} - e^{x} \]antiderivativeEvaluate the final integral.✓ Proved
Answer \( \left(x - 1\right) e^{x} + C \)
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. 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 |
| 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) — The solution applies more than one rule in a single step and misuses integration by parts. Step 2 adds and subtracts the same term in one line, violating the one‑change rule, and step 3 incorrectly rewrites the integral as a difference of two integrals. Step 6 again applies integration by parts to a trivial integral, which is unnecessary and incorrect. The algebraic manipulations do not correctly follow the integration‑by‑parts formula, leading to an invalid derivation.qwen3.6:27b-mlx: fail (error) — The solution incorrectly applies integration by parts in steps 2-3, resulting in the tautology Integral(x*exp(x), x) = Integral(x*exp(x), x). Step 6 then magically produces the correct boundary term x*exp(x) without a valid preceding step, breaking the logical chain.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly applies integration by parts in steps 2-3, resulting in the tautology Integral(x*exp(x), x) = Integral(x*exp(x), x). Step 6 then magically produces the correct boundary term x*exp(x) without a valid preceding step, breaking the logical chain.gpt-oss:20b: fail (error) 2026-10-04 — The solution applies more than one rule in a single step and misuses integration by parts. Step 2 adds and subtracts the same term in one line, violating the one‑change rule, and step 3 incorrectly rewrites the integral as a difference of two integrals. Step 6 again applies integration by parts to a trivial integral, which is unnecessary and incorrect. The algebraic manipulations do not correctly follow the integration‑by‑parts formula, leading to an invalid derivation.qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "Step 2 is mathematically invalid; adding and subtracting terms inside the integral does not constitute integration by parts, and the transition to Stgpt-oss:20b: fail (style) 2026-10-04 — Steps 2 and 6 incorrectly label the operations. Step 2 merely adds a zero term and does not perform integration by parts; the correct label would be "rewrite" or "simplify". Step 6 applies the "parts" rule to the integral of e^x, which is unnecessary and misleading—this step should be labeled "integral" or "antiderivative" instead.
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 2026-10-04 with SymPy 1.14.0.