Home›Calculus 2›Integration by parts›Problem 4.1407

Integral of \( \displaystyle \left(2 x + 1\right) e^{- 2 x - 1} \)

Problem 4.1407 · medium Video

Find \( \displaystyle \int \left(2 x + 1\right) e^{- 2 x - 1} \, dx \). (Omit the constant of integration.)

\[ \int \left(2 x + 1\right) e^{- 2 x - 1}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(2 x e^{- 2 x - 1} + e^{- 2 x - 1}\right)\, dx \]
linearityDistribute the exponential term.✓ Proved
\[ = 2 \int x e^{- 2 x - 1}\, dx + \int e^{- 2 x - 1}\, dx \]
linearitySplit into two integrals.✓ Proved
\[ = - x e^{- 2 x - 1} - 2 \int \left(- \frac{e^{- 2 x - 1}}{2}\right)\, dx + \int e^{- 2 x - 1}\, dx \]
partsApply integration by parts to the first integral with u=x and dv=exp(-2*x-1)dx.✓ Proved
\[ = - x e^{- 2 x - 1} + 2 \int e^{- 2 x - 1}\, dx \]
algebra algebra algebraSimplify the sign inside the parentheses. Distribute the constant 2. Combine the integral terms.✓ Proved
\[ = - x e^{- 2 x - 1} - e^{- 2 x - 1} \]
antiderivative algebraEvaluate the remaining integral. Simplify the final expression.✓ Proved

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

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

✓ 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. 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
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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-02
gpt-oss:20b: pass 2026-11-02
qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly applies integration by parts and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly applies integration by parts and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-11-01

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 2026-11-02 with SymPy 1.14.0.