Integral of \( \displaystyle \left(3 x - 1\right) \cos{\left(3 x - 1 \right)} \)
Problem 4.804 · medium
Find \( \displaystyle \int \left(3 x - 1\right) \cos{\left(3 x - 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(3 x - 1\right) \cos{\left(3 x - 1 \right)}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = \frac{\left(3 x - 1\right) \sin{\left(3 x - 1 \right)}}{3} - \int \sin{\left(3 x - 1 \right)}\, dx \]parts simplifyApply integration by parts with u = 3*x - 1 and dv = cos(3*x - 1) dx. Simplify the integrand in the remaining integral.✓ Proved
- \[ = \frac{\left(3 x - 1\right) \sin{\left(3 x - 1 \right)}}{3} + \frac{\cos{\left(3 x - 1 \right)}}{3} \]antiderivative simplifyIntegrate the remaining sine term. Simplify the final expression.✓ Proved
Answer \( x \sin{\left(3 x - 1 \right)} - \frac{\sin{\left(3 x - 1 \right)}}{3} + \frac{\cos{\left(3 x - 1 \right)}}{3} + C \)
✓ 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.
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 |
| 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: passqwen3.6:27b-mlx: pass — The solution correctly applies integration by parts, simplifies the resulting integrand, and computes the final antiderivative. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies integration by parts, simplifies the resulting integrand, and computes the final antiderivative. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies integration by parts, simplifies the resulting integrand, and computes the final antiderivative. Each step adheres to the one-rule-per-step constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-10
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 2026-10-10 with SymPy 1.14.0.