Integral of \( \displaystyle \sin^{2}{\left(3 x \right)} \)
Problem 4.182 · easy Video
Find \( \displaystyle \int \sin^{2}{\left(3 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sin^{2}{\left(3 x \right)}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = \int \left(\frac{1}{2} - \frac{\cos{\left(6 x \right)}}{2}\right)\, dx \]trig-identityUse the power-reduction identity sin(theta)^2 = (1 - cos(2*theta))/2.✓ Proved
- \[ = \int \frac{1}{2}\, dx - \int \frac{\cos{\left(6 x \right)}}{2}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = \frac{x}{2} - \int \frac{\cos{\left(6 x \right)}}{2}\, dx \]antiderivativeIntegrate the constant term.✓ Proved
- \[ = \frac{x}{2} - \frac{\sin{\left(6 x \right)}}{12} \]antiderivativeIntegrate the cosine term.✓ Proved
Answer \( \frac{x}{2} - \frac{\sin{\left(6 x \right)}}{12} + C \)
✓ 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. 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 |
| 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
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29gpt-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.