Home›Calculus 2›Integration by parts›Problem 4.1353

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

Problem 4.1353 · 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
\[ = \int \left(x e^{- x - 1} + e^{- x - 1}\right)\, dx \]
linearityDistribute the exponential term.✓ Proved
\[ = \int x e^{- x - 1}\, dx + \int e^{- x - 1}\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = - e^{- x - 1} + \int x e^{- x - 1}\, dx \]
antiderivative simplifyEvaluate the second integral. Simplify the expression.✓ Proved
\[ = \operatorname{parts}{\left(x,e^{- x - 1},1,- e^{- x - 1} \right)} - e^{- x - 1} \]
partsApply integration by parts to the first integral.Reviewed
\[ = - x e^{- x - 1} - e^{- x - 1} - \int \left(- e^{- x - 1}\right)\, dx \]
partsApply the integration by parts formula.Reviewed
\[ = - x e^{- x - 1} - e^{- x - 1} + \int e^{- x - 1}\, dx \]
algebraDistribute the negative sign into the integral.✓ Proved
\[ = - x e^{- x - 1} - 2 e^{- x - 1} \]
antiderivative simplifyEvaluate the remaining integral. Combine 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).

Lines: 9 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	simplify left -(x + parts(x, exp(-x - 1), 1, -exp(-x - 1))exp(x + 1) + 1)exp(-x - 1); no point in the sample was defined on both lines
7	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	simplify left (x + parts(x, exp(-x - 1), 1, -exp(-x - 1))exp(x + 1) + 1)exp(-x - 1); 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
10	✓ 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: pass — The solution correctly applies integration by parts and linearity. Each step changes only one aspect of the expression, and the labels accurately describe the operations performed.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-27 — The solution correctly applies integration by parts and linearity. Each step changes only one aspect of the expression, and the labels accurately describe the operations performed.
gpt-oss:20b: pass 2026-10-27
qwen3.6:27b-mlx: fail (style) 2026-10-26 — Step 6 violates the one-rule-per-step constraint by simultaneously performing the integration by parts setup and the subtraction of the second term from Step 5. Step 7 also incorrectly labels the expansion of the parts formula as 'parts' when it is an algebraic expansion.
gpt-oss:20b: pass 2026-10-26
qwen3.6:27b-mlx: pass 2026-10-26
gpt-oss:20b: pass 2026-10-26

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 2026-10-27 with SymPy 1.14.0.