Home›Calculus 2›Integration by parts›Problem 4.1320

Integral of \( \displaystyle \left(2 x + 1\right) \sin{\left(2 x + 1 \right)} \)

Problem 4.1320 · medium Video

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

\[ \int \left(2 x + 1\right) \sin{\left(2 x + 1 \right)}\, dx \]
integralStart with the integral of the integrand.✓ Proved
\[ = - \frac{\left(2 x + 1\right) \cos{\left(2 x + 1 \right)}}{2} - \int \left(- \cos{\left(2 x + 1 \right)}\right)\, dx \]
parts algebraApply integration by parts with u = 2*x + 1 and dv = sin(2*x + 1) dx. Simplify the integrand in the remaining integral.✓ Proved
\[ = - \frac{\left(2 x + 1\right) \cos{\left(2 x + 1 \right)}}{2} + \int \cos{\left(2 x + 1 \right)}\, dx \]
algebraDistribute the negative sign.✓ Proved
\[ = - \frac{\left(2 x + 1\right) \cos{\left(2 x + 1 \right)}}{2} + \frac{\sin{\left(2 x + 1 \right)}}{2} \]
antiderivativeIntegrate the remaining cosine term.✓ Proved
\[ = \frac{\left(- 2 x - 1\right) \cos{\left(2 x + 1 \right)}}{2} + \frac{\sin{\left(2 x + 1 \right)}}{2} \]
algebraDistribute the fraction.✓ Proved
\[ = - x \cos{\left(2 x + 1 \right)} + \frac{\sin{\left(2 x + 1 \right)}}{2} - \frac{\cos{\left(2 x + 1 \right)}}{2} \]
algebraExpand the expression.✓ Proved

Answer \( - x \cos{\left(2 x + 1 \right)} + \frac{\sin{\left(2 x + 1 \right)}}{2} - \frac{\cos{\left(2 x + 1 \right)}}{2} + C \)

Top: the graph of F = -x·cos(2x + 1) + sin(2x + 1)/2 - cos(2x + 1)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x + 1)·sin(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
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 subsequent algebraic simplifications. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-23 — The solution correctly applies integration by parts and subsequent algebraic simplifications. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-23
qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly applies integration by parts and subsequent algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-22
qwen3.6:27b-mlx: pass 2026-10-22
gpt-oss:20b: pass 2026-10-22

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 2026-10-23 with SymPy 1.14.0.