Integral of \( \displaystyle 3 x \cos{\left(6 x \right)} \)
Problem 4.150 · medium Video
Find \( \displaystyle \int 3 x \cos{\left(6 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int 3 x \cos{\left(6 x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = 3 \int x \cos{\left(6 x \right)}\, dx \]linearityFactor out the constant 3.✓ Proved
- \[ = \frac{x \sin{\left(6 x \right)}}{2} - 3 \int \frac{\sin{\left(6 x \right)}}{6}\, dx \]parts algebraApply integration by parts with u = x and dv = cos(6*x)dx. Distribute the 3 into the parentheses.✓ Proved
- \[ = \frac{x \sin{\left(6 x \right)}}{2} + \frac{\cos{\left(6 x \right)}}{12} \]antiderivative algebra simplifyEvaluate the integral of sin(6*x)/6. Simplify the signs and fractions. Distribute the 3 and simplify the final expression.✓ Proved
Answer \( \frac{x \sin{\left(6 x \right)}}{2} + \frac{\cos{\left(6 x \right)}}{12} + C \)
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. 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 | ✓ 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 |
| 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 claims to distribute the factor 3 into the parentheses, but the expression remains unchanged. The step applies no change, violating the rule that each step must alter exactly one element.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (error) 2026-10-03 — Step 4 claims to distribute the factor 3 into the parentheses, but the expression remains unchanged. The step applies no change, violating the rule that each step must alter exactly one element.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies integration by parts and algebraic simplification. 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: fail (style) 2026-09-29 — Step 4 is labeled 'algebra' but performs no algebraic operation; it merely removes redundant parentheses. Step 7 is labeled 'simplify' but performs distribution, which is an algebraic operation.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.