Integral of \( \displaystyle \left(3 x - 1\right) \cos{\left(6 x - 2 \right)} \)
Problem 4.929 · medium
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 function.✓ Proved
- \[ = \int 3 x \cos{\left(6 x - 2 \right)}\, dx - \int \cos{\left(6 x - 2 \right)}\, dx \]linearityDistribute the terms.✓ Proved
- \[ = 3 \int x \cos{\left(6 x - 2 \right)}\, dx - \int \cos{\left(6 x - 2 \right)}\, dx \]algebra partsFactor out the constant 3. This is not the correct way to show parts. Let's use the standard parts formula.✓ Proved
- \[ = \frac{x \sin{\left(6 x - 2 \right)}}{2} - 3 \int \frac{\sin{\left(6 x - 2 \right)}}{6}\, dx - \int \cos{\left(6 x - 2 \right)}\, dx \]parts algebraApply integration by parts to the first integral term. Distribute the 3.✓ Proved
- \[ = \frac{x \sin{\left(6 x - 2 \right)}}{2} - \frac{\int \sin{\left(6 x - 2 \right)}\, dx}{2} - \int \cos{\left(6 x - 2 \right)}\, dx \]algebraSimplify the coefficients.✓ 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 algebraEvaluate the remaining integrals. Simplify the expression.✓ Proved
- \[ = \left(\frac{x}{2} - \frac{1}{6}\right) \sin{\left(6 x - 2 \right)} + \frac{\cos{\left(6 x - 2 \right)}}{12} \]algebraGroup the sine terms.✓ Proved
- \[ = \frac{\left(3 x - 1\right) \sin{\left(6 x - 2 \right)}}{6} + \frac{\cos{\left(6 x - 2 \right)}}{12} \]simplifyFinal simplified form.✓ Proved
Answer \( \frac{x \sin{\left(6 x - 2 \right)}}{2} - \frac{\sin{\left(6 x - 2 \right)}}{6} + \frac{\cos{\left(6 x - 2 \right)}}{12} + C \)
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. A reviewer's objection to the explanation stands; see the receipt.
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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ 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 4 applies two operations at once: it attempts to perform integration by parts and algebraic manipulation simultaneously, and it introduces an incorrect equality sign. The step should separate the integration by parts from the algebraic simplification and avoid the spurious "== 0".qwen3.6:27b-mlx: fail (error) — Step 4 is mathematically nonsensical and syntactically invalid, containing a self-referential equality check ('== 0') and failing to actually perform the integration by parts operation. Step 5 then abruptly presents the correct result of the parts operation without a valid transition from Step 4.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — Step 4 applies two operations at once: it attempts to perform integration by parts and algebraic manipulation simultaneously, and it introduces an incorrect equality sign. The step should separate the integration by parts from the algebraic simplification and avoid the spurious "== 0".qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 4 is mathematically nonsensical and syntactically invalid, containing a self-referential equality check ('== 0') and failing to actually perform the integration by parts operation. Step 5 then abruptly presents the correct result of the parts operation without a valid transition from Step 4.qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 4 is mathematically nonsensical, containing a self-referential equality 'Integral(...) == 0' and failing to correctly apply integration by parts. Step 5 then magically produces the correct result from this garbage without showing the actual integration of the resulting term, violating the one-step rule and logical continuity.gpt-oss:20b: fail (error) 2026-10-12 — Step 4 is incorrect: it mixes an equality sign inside an expression and applies the integration‑by‑parts formula incorrectly, effectively applying more than one rule at once and producing an invalid intermediate expression.
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 2026-10-12 with SymPy 1.14.0.