Integral of \( \displaystyle \sin{\left(6 x \right)} \cos{\left(3 x \right)} \)
Problem 4.478 · medium
Find \( \displaystyle \int \sin{\left(6 x \right)} \cos{\left(3 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sin{\left(6 x \right)} \cos{\left(3 x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int 2 \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}\, dx \]trig-identity algebraUse the double angle identity sin(2a) = 2sin(a)cos(a) for 6x = 2(3x). Simplify the expression.✓ Proved
- \[ = 2 \int \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}\, dx \]algebra algebraFactor out the constant. Rearrange the terms.✓ Proved
- \[ = - \frac{2 \cos^{3}{\left(3 x \right)}}{9} \]antiderivative simplifyUse substitution u = cos(3*x), du = -3sin(3*x)dx. Simplify the final expression.✓ Proved
Answer \( - \frac{2 \cos^{3}{\left(3 x \right)}}{9} + C \)
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.
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 (style) — Step 6 applies both a substitution (u=cos(3x)) and an antiderivative in a single step, violating the one‑rule‑per‑step rule. The step should be split into two separate steps, one for substitution and one for integration.qwen3.6:27b-mlx: fail (error) — Step 6 applies the antiderivative rule but fails to account for the chain rule factor of 1/3 from the inner derivative of cos(3x), resulting in an incorrect coefficient. The step claims to use substitution but produces a mathematically wrong result.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-06 — Step 6 applies both a substitution (u=cos(3x)) and an antiderivative in a single step, violating the one‑rule‑per‑step rule. The step should be split into two separate steps, one for substitution and one for integration.qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 6 applies the antiderivative rule but fails to account for the chain rule factor of 1/3 from the inner derivative of cos(3x), resulting in an incorrect coefficient. The step claims to use substitution but produces a mathematically wrong result.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 6 claims to use the rule 'antiderivative' but the note describes a substitution process. The step skips the explicit substitution step (changing variables) and the integration step, applying multiple logical operations (substitution, integration, and back-substitution) in a single line, which violates the 'one rule per step' constraint. Additionally, the label 'antiderivative' is not in the allowed vocabulary list for this specific operation type (usually 'substitution' or 'integral' depending on the system's granularity, but here the main issue is the multi-step compression).
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 2026-10-06 with SymPy 1.14.0.