Integral of \( \displaystyle \cos{\left(6 x - 2 \right)} \)
Problem 4.871 · easy
Find \( \displaystyle \int \cos{\left(6 x - 2 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \cos{\left(6 x - 2 \right)}\, dx \]integral rewrite simplifyStart with the integral of the function. Factor out the 6 from the argument. Revert to the original form to prepare for integration.✓ Proved
- \[ = \frac{\sin{\left(6 x - 2 \right)}}{6} \]antiderivativeApply the rule for the antiderivative of cos(ax + b).✓ Proved
Answer \( \frac{\sin{\left(6 x - 2 \right)}}{6} + C \)
✓ 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 |
| 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 identifies the antiderivative using standard rules. The intermediate steps involving rewriting and simplifying the argument are valid presentational steps that do not violate the single-rule constraint.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the antiderivative using standard rules. The intermediate steps involving rewriting and simplifying the argument are valid presentational steps that do not violate the single-rule constraint.qwen3.6:27b-mlx: pass 2026-10-11 — The solution is correct and follows the allowed rules. The intermediate steps (rewriting and simplifying) are redundant but not defective under the contract, as they apply single rules correctly.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.