Home›Calculus 2›Integration by parts›Problem 4.163

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

Problem 4.163 · medium Video

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

\[ \int \left(x + 1\right) \cos{\left(x + 1 \right)}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(x \cos{\left(x + 1 \right)} + \cos{\left(x + 1 \right)}\right)\, dx \]
algebraDistribute the term (x + 1) into the cosine.✓ Proved
\[ = \int x \cos{\left(x + 1 \right)}\, dx + \int \cos{\left(x + 1 \right)}\, dx \]
linearitySplit the integral into two separate integrals.✓ Proved
\[ = \sin{\left(x + 1 \right)} + \int x \cos{\left(x + 1 \right)}\, dx \]
antiderivativeIntegrate the second term cos(x + 1).✓ Proved
\[ = x \sin{\left(x + 1 \right)} + \sin{\left(x + 1 \right)} - \int \sin{\left(x + 1 \right)}\, dx \]
partsApply integration by parts to the first term with u = x and dv = cos(x + 1)dx.✓ Proved
\[ = x \sin{\left(x + 1 \right)} + \sin{\left(x + 1 \right)} + \cos{\left(x + 1 \right)} \]
antiderivative algebraIntegrate sin(x + 1). Simplify the signs.✓ Proved
\[ = \left(x + 1\right) \sin{\left(x + 1 \right)} + \cos{\left(x + 1 \right)} \]
algebraFactor out sin(x + 1) to get the final form.✓ Proved

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

Top: the graph of F = x·sin(x + 1) + sin(x + 1) + cos(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x + 1)·cos(x + 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
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 linearity. 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 linearity. 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
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — 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 2026-09-29

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.