Integral of \( \displaystyle \sin^{2}{\left(3 x - 1 \right)} \)
Problem 4.228 · medium
Find \( \displaystyle \int \sin^{2}{\left(3 x - 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sin^{2}{\left(3 x - 1 \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(\frac{1}{2} - \frac{\cos{\left(6 x - 2 \right)}}{2}\right)\, dx \]trig-identityUse the power-reduction identity sin(u)^2 = (1 - cos(2u))/2.✓ Proved
- \[ = \int \frac{1}{2}\, dx - \int \frac{\cos{\left(6 x - 2 \right)}}{2}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = \frac{x}{2} - \int \frac{\cos{\left(6 x - 2 \right)}}{2}\, dx \]antiderivativeIntegrate the first constant term.✓ Proved
- \[ = \frac{x}{2} - \frac{\sin{\left(6 x - 2 \right)}}{12} \]antiderivative simplifyIntegrate the cosine term using the substitution rule for linear arguments. Simplify the final expression.✓ Proved
Answer \( \frac{x}{2} - \frac{\sin{\left(6 x - 2 \right)}}{12} + C \)
✓ 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.
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 |
| 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 the power-reduction identity, linearity of integration, and standard antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the allowed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the power-reduction identity, linearity of integration, and standard antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the allowed vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the power-reduction identity, linearity of integration, and standard antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the allowed vocabulary.gpt-oss:20b: pass 2026-10-03
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.