Integral of \( \displaystyle \sin{\left(3 x \right)} + \cos{\left(3 x \right)} \)
Problem 4.472 · medium
Find \( \displaystyle \int \sin{\left(3 x \right)} + \cos{\left(3 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right)\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \sin{\left(3 x \right)}\, dx + \int \cos{\left(3 x \right)}\, dx \]linearitySplit the integral into two parts using linearity.✓ Proved
- \[ = \frac{\sin{\left(3 x \right)}}{3} - \frac{\cos{\left(3 x \right)}}{3} \]antiderivativeIntegrate each term separately using the rule for sin(ax) and cos(ax).✓ Proved
Answer \( \frac{\sin{\left(3 x \right)} - \cos{\left(3 x \right)}}{3} + C \)
✓ 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 |
| 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 linearity to split the integral and then applies the antiderivative rule to each term. The final expression is algebraically equivalent to the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies linearity to split the integral and then applies the antiderivative rule to each term. The final expression is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (style) 2026-10-06 — Step 3 applies the antiderivative rule to two separate terms simultaneously, violating the contract that each step must change only one thing.
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.