Home›Calculus 2›Integration by parts›Problem 4.140

Integral of \( \displaystyle x \cos{\left(2 x \right)} \)

Problem 4.140 · easy Video

Find \( \displaystyle \int x \cos{\left(2 x \right)} \, dx \). (Omit the constant of integration.)

\[ \int x \cos{\left(2 x \right)}\, dx \]
integralStart with the integral of the integrand.✓ Proved
\[ = \operatorname{parts}{\left(x \cos{\left(2 x \right)},x,1,\cos{\left(2 x \right)},-1 \right)} \]
partsApply integration by parts where u = x and dv = cos(2*x)dx.Reviewed
\[ = \frac{x \sin{\left(2 x \right)}}{2} - \int \frac{\sin{\left(2 x \right)}}{2}\, dx \]
parts algebraApply the integration by parts formula: uv - Integral(v du). Simplify the expression inside the integral.Reviewed
\[ = \frac{x \sin{\left(2 x \right)}}{2} - \frac{\int \sin{\left(2 x \right)}\, dx}{2} \]
linearityFactor out the constant 1/2 from the integral.✓ Proved
\[ = \frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4} \]
antiderivative algebraEvaluate the integral of sin(2*x). Simplify the signs and coefficients.✓ Proved

Answer \( \frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4} + C \)

Top: the graph of F = x·sin(2x)/2 + cos(2x)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = x·cos(2x).

Lines: 6 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	simplify left xsin(2x)/2 - parts(xcos(2x), x, 1, cos(2x), -1) + cos(2x)/4; no point in the sample was defined on both lines
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	simplify left -xsin(2x)/2 + parts(xcos(2x), x, 1, cos(2x), -1) - cos(2x)/4; no point in the sample was defined on both lines
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: pass — The solution correctly applies integration by parts and standard antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies integration by parts and standard antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies integration by parts and standard antiderivative rules. 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
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.