Integral of \( \displaystyle 3 x \sin{\left(3 x \right)} \)
Problem 4.1333 · medium Video
Find \( \displaystyle \int 3 x \sin{\left(3 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int 3 x \sin{\left(3 x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \operatorname{parts}{\left(3 x \sin{\left(3 x \right)},x,1,1,3 x,\sin{\left(3 x \right)} \right)} \]partsUse integration by parts with u = 3*x and dv = sin(3*x)dx.Not checked
- \[ = - x \cos{\left(3 x \right)} + \int \cos{\left(3 x \right)}\, dx \]partsApply the integration by parts formula: integral(u dv) = u*v - integral(v du).Not checked
- \[ = - x \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{3} \]antiderivativeEvaluate the remaining integral.✓ Proved
Answer \( - x \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{3} + C \)
Lines: 3 proved, 2 not checked. 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 | Not checked | — | simplify left -x*cos(3*x) - parts(3*x*sin(3*x), x, 1, 1, 3*x, sin(3*x)) + sin(3*x)/3; no point in the sample was defined on both lines |
| 3 | Not checked | — | simplify left x*cos(3*x) + parts(3*x*sin(3*x), x, 1, 1, 3*x, sin(3*x)) - sin(3*x)/3; no point in the sample was defined on both lines |
| 4 | ✓ 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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 2 incorrectly applies integration by parts: it takes \(v=\sin(3x)\) instead of the correct antiderivative \(-\cos(3x)/3\). Consequently, the subsequeqwen3.6:27b-mlx: pass — The solution correctly applies integration by parts and evaluates the remaining integral. 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 2026-10-24 — The solution correctly applies integration by parts and evaluates the remaining integral. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: inconclusive 2026-10-24 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 2 incorrectly applies integration by parts: it takes \(v=\sin(3x)\) instead of the correct antiderivative \(-\cos(3x)/3\). Consequently, the subsequeqwen3.6:27b-mlx: pass 2026-10-23 — The solution correctly applies integration by parts and evaluates the remaining integral. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-23qwen3.6:27b-mlx: pass 2026-10-23 — The solution correctly applies integration by parts and evaluates the remaining integral. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-23
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 2026-10-24 with SymPy 1.14.0.