Integral of \( \displaystyle x \sin{\left(x \right)} \)
Problem 4.142 · easy Video
Find \( \displaystyle \int x \sin{\left(x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int x \sin{\left(x \right)}\, dx \]integralStart with the integral of the integrand.✓ Proved
- \[ = \int \left(- x \frac{d}{d x} \cos{\left(x \right)}\right)\, dx \]rewriteRewrite sin(x) as the negative derivative of cos(x).✓ Proved
- \[ = - x \cos{\left(x \right)} - \int \left(- \cos{\left(x \right)} \frac{d}{d x} x\right)\, dx \]partsApply integration by parts with u = x and dv = -cos(x)dx.✓ Proved
- \[ = - x \cos{\left(x \right)} - \int \left(- \cos{\left(x \right)}\right)\, dx \]simplifySimplify the derivative of x.✓ Proved
- \[ = - x \cos{\left(x \right)} + \int \cos{\left(x \right)}\, dx \]algebraDistribute the negative sign into the integral.✓ Proved
- \[ = - x \cos{\left(x \right)} + \sin{\left(x \right)} \]antiderivativeEvaluate the remaining integral.✓ Proved
Answer \( - x \cos{\left(x \right)} + \sin{\left(x \right)} + C \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt. 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: fail (error) — Step 3 applies integration by parts incorrectly: with u=x and dv=-cos(x)dx, the product uv should be x*(-sin(x)), not x*(-cos(x)). The subsequent sign manipulations in steps 4–5 also do not follow from the correct application of the rule, leading to an incorrect antiderivative.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (error) 2026-10-03 — Step 3 applies integration by parts incorrectly: with u=x and dv=-cos(x)dx, the product uv should be x*(-sin(x)), not x*(-cos(x)). The subsequent sign manipulations in steps 4–5 also do not follow from the correct application of the rule, leading to an incorrect antiderivative.qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: fail (error) 2026-09-29 — Step 3 applies integration by parts incorrectly: with u=x and dv=-cos(x)dx, the antiderivative v should be -sin(x), not -cos(x). The subsequent steps rely on this wrong choice, making the solution mathematically incorrect.qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: fail (error) 2026-09-29 — Step 3 applies integration by parts incorrectly: with u=x and dv=-cos(x)dx, the antiderivative v should be -sin(x), not -cos(x). The resulting expression x*(-cos(x)) is algebraically incorrect and combines two errors in one step.
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.