Integral of \( \displaystyle \left(x - 1\right) e^{1 - x} \)
Problem 4.159 · medium Video
Find \( \displaystyle \int \left(x - 1\right) e^{1 - x} \, dx \). (Omit the constant of integration.)
- \[ \int \left(x - 1\right) e^{1 - x}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = \int \left(x e^{1 - x} - e^{1 - x}\right)\, dx \]linearityDistribute the exponential term.✓ Proved
- \[ = \int x e^{1 - x}\, dx - \int e^{1 - x}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = e^{1 - x} + \int x e^{1 - x}\, dx \]antiderivative algebraIntegrate the second term. Simplify the signs.✓ Proved
- \[ = \operatorname{parts}{\left(x e^{1 - x},x,1,e^{1 - x},- e^{1 - x},0 \right)} \]partsUse integration by parts on the first integral.Not checked
- \[ = - x e^{1 - x} + e^{1 - x} + \int e^{1 - x}\, dx \]partsApply the integration by parts formula.Not checked
- \[ = - x e^{1 - x} \]antiderivative simplifyIntegrate the remaining exponential term. Combine the remaining terms.✓ Proved
Answer \( - x e^{1 - x} + C \)
Lines: 8 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt. 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 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | Not checked | — | simplify left (-x - parts(x*exp(1 - x), x, 1, exp(1 - x), -exp(1 - x), 0)*exp(x - 1))*exp(1 - x); no point in the sample was defined on both lines |
| 7 | Not checked | — | simplify left (x + parts(x*exp(1 - x), x, 1, exp(1 - x), -exp(1 - x), 0)*exp(x - 1))*exp(1 - x); no point in the sample was defined on both lines |
| 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 7 incorrectly adds an extra +exp(1‑x); the integration‑by‑parts result should be –x·exp(1‑x)+∫exp(1‑x)dx, not –x·exp(1‑x)+∫exp(1‑x)dx+exp(1‑x).qwen3.6:27b-mlx: fail (error) — Step 6 is not a valid transformation step; it is a meta-commentary or annotation describing the choice of u and dv for integration by parts, rather than an algebraic or calculus operation that transforms the expression in Step 5 into Step 7. Furthermore, Step 7 applies the integration by parts formula, but the label 'parts' is applied to the result of the operation, whereas the transition from 5 to 7 (skipping the invalid step 6) would be the application of 'parts'. The inclusion of Step 6 as a distinct line in the derivation chain violates the requirement that each step changes the expression via a named rule.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 6 is not a valid transformation step; it is a meta-commentary or annotation describing the choice of u and dv for integration by parts, rather than an algebraic or calculus operation that transforms the expression in Step 5 into Step 7. Furthermore, Step 7 applies the integration by parts formula, but the label 'parts' is applied to the result of the operation, whereas the transition from 5 to 7 (skipping the invalid step 6) would be the application of 'parts'. The inclusion of Step 6 as a distinct line in the derivation chain violates the requirement that each step changes the expression via a named rule.gpt-oss:20b: fail (error) 2026-10-03 — Step 7 incorrectly adds an extra +exp(1‑x); the integration‑by‑parts result should be –x·exp(1‑x)+∫exp(1‑x)dx, not –x·exp(1‑x)+∫exp(1‑x)dx+exp(1‑x).qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: fail (error) 2026-09-29 — Step 6 incorrectly applies integration by parts: it sets u=1 and dv=exp(1‑x) dx, but then assigns v=0, which is inconsistent with the rule. The correct choice should be u=x, dv=exp(1‑x) dx, yielding du=dx and v=−exp(1‑x).qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: fail (error) 2026-09-29 — Step 6 attempts to invoke the integration‑by‑parts rule but does not actually apply it to the integral; it merely writes a function call with incorrect arguments, violating the rule‑application granularity requirement.
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.