Home›Calculus 2›Integration by parts›Problem 4.145

Integral of \( \displaystyle \left(x - 1\right) e^{x - 1} \)

Problem 4.145 · medium Video

Find \( \displaystyle \int \left(x - 1\right) e^{x - 1} \, dx \). (Omit the constant of integration.)

\[ \int \left(x - 1\right) e^{x - 1}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(x e^{x - 1} - e^{x - 1}\right)\, dx \]
algebraDistribute the exponential term.✓ Proved
\[ = \int x e^{x - 1}\, dx - \int e^{x - 1}\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = - e^{x - 1} + \int x e^{x - 1}\, dx \]
antiderivativeIntegrate the second term.✓ Proved
\[ = x e^{x - 1} - e^{x - 1} - \int e^{x - 1}\, dx \]
rewriteRewrite the integrand of the first term for integration by parts.✓ Proved
\[ = x e^{x - 1} - 2 e^{x - 1} \]
parts simplifyApply integration by parts to the first integral. Combine the like terms.✓ Proved

Answer \( \left(x - 2\right) e^{x - 1} + C \)

Top: the graph of F = (x - 2)·e^(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x - 1)·e^(x - 1).

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. 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	✓ 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: pass
qwen3.6:27b-mlx: fail (error) — Step 5 is not a valid algebraic rewrite; it incorrectly transforms an unevaluated integral into a difference of a term and another integral without justification. Step 6 then claims to apply integration by parts, but the structure of the expression does not match the standard application of the rule to the integral in Step 4.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 is not a valid algebraic rewrite; it incorrectly transforms an unevaluated integral into a difference of a term and another integral without justification. Step 6 then claims to apply integration by parts, but the structure of the expression does not match the standard application of the rule to the integral in Step 4.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 5 is mathematically invalid; it replaces an unevaluated integral with a term minus another integral without justification, violating the single-rule constraint and logical flow. Step 6 then incorrectly claims to apply integration by parts to a term that was already partially resolved in the previous erroneous step.
gpt-oss:20b: fail (style) 2026-09-29 — Step 5 incorrectly labels the application of integration by parts as "rewrite"; it should be labeled "parts".
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 5 is mathematically invalid; it asserts that the unevaluated integral Integral(x*exp(x - 1), x) is equal to the expression x*exp(x - 1) - Integral(1*exp(x - 1), x) without performing the integration by parts operation. This conflates the setup for integration by parts with the result, violating the requirement that steps be algebraically equal. Step 6 then attempts to evaluate the remaining integral, but the previous step already incorrectly resolved the first term.
gpt-oss:20b: pass 2026-09-29

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.