Integral of \( \displaystyle \left(3 x + 1\right) \cos{\left(6 x + 2 \right)} \)
Problem 4.153 · medium Video
Find \( \displaystyle \int \left(3 x + 1\right) \cos{\left(6 x + 2 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(3 x + 1\right) \cos{\left(6 x + 2 \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(3 x \cos{\left(6 x + 2 \right)} + \cos{\left(6 x + 2 \right)}\right)\, dx \]linearityDistribute the cosine term.✓ Proved
- \[ = 3 \int x \cos{\left(6 x + 2 \right)}\, dx + \int \cos{\left(6 x + 2 \right)}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = \frac{x \sin{\left(6 x + 2 \right)}}{2} - 3 \int \frac{\sin{\left(6 x + 2 \right)}}{6}\, dx + \int \cos{\left(6 x + 2 \right)}\, dx \]partsApply integration by parts to the first integral with u=x and dv=cos(6x+2)dx.✓ Proved
- \[ = \frac{x \sin{\left(6 x + 2 \right)}}{2} + \frac{\sin{\left(6 x + 2 \right)}}{6} - 3 \int \frac{\sin{\left(6 x + 2 \right)}}{6}\, dx \]antiderivativeEvaluate the integral of cos(6x+2).✓ Proved
- \[ = \frac{x \sin{\left(6 x + 2 \right)}}{2} + \frac{\sin{\left(6 x + 2 \right)}}{6} + \frac{\cos{\left(6 x + 2 \right)}}{12} \]antiderivative simplify simplifyEvaluate the integral of sin(6x+2). Simplify the expression inside the parentheses. Distribute the 3 and simplify.✓ Proved
- \[ = \left(\frac{x}{2} + \frac{1}{6}\right) \sin{\left(6 x + 2 \right)} + \frac{\cos{\left(6 x + 2 \right)}}{12} \]simplifyCombine the sine terms.✓ Proved
Answer \( \frac{x \sin{\left(6 x + 2 \right)}}{2} + \frac{\sin{\left(6 x + 2 \right)}}{6} + \frac{\cos{\left(6 x + 2 \right)}}{12} + C \)
✓ 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. 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 |
| 9 | ✓ 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 and linearity. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
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 the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-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 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies linearity, 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.