Home›Calculus 2›Integration by parts›Problem 4.160

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

Problem 4.160 · medium Video

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

\[ \int 2 x e^{2 x}\, dx \]
integralStart with the integral of the function.✓ Proved
\[ = x e^{2 x} - \int e^{2 x}\, dx \]
parts simplifyApply integration by parts with u = 2*x and dv = exp(2*x) dx. Simplify the terms inside and outside the integral.✓ Proved
\[ = x e^{2 x} - \frac{e^{2 x}}{2} \]
antiderivative simplifyEvaluate the remaining integral. Final simplified form of the antiderivative.✓ Proved

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

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

✓ 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
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 and simplifies the result. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies integration by parts and simplifies the result. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
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.