Home›Calculus 2›Integration by parts›Problem 4.144

Integral of \( \displaystyle x^{2} e^{x} \)

Problem 4.144 · easy Video

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

\[ \int x^{2} e^{x}\, dx \]
integralStart with the integral of the integrand.✓ Proved
\[ = x^{2} e^{x} - \int 2 x e^{x}\, dx \]
partsApply integration by parts with u = x**2 and dv = exp(x) dx.✓ Proved
\[ = x^{2} e^{x} - 2 \int x e^{x}\, dx \]
linearityPull the constant factor out of the integral.✓ Proved
\[ = x^{2} e^{x} - 2 x e^{x} + 2 \int e^{x}\, dx \]
partsApply integration by parts to the remaining integral with u = x and dv = exp(x) dx.✓ Proved
\[ = x^{2} e^{x} - 2 x e^{x} + 2 e^{x} \]
antiderivative algebraEvaluate the integral of exp(x). Distribute the constant and simplify.✓ Proved

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

Top: the graph of F = (x² - 2x + 2)·e^(x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = x²·e^(x).

✓ 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
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 2026-10-03
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies integration by parts twice, followed by standard algebraic simplification. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29
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.