Integral of \( \displaystyle \sin^{3}{\left(3 x \right)} \)
Problem 4.1264 · easy Video
Find \( \displaystyle \int \sin^{3}{\left(3 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sin^{3}{\left(3 x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(1 - \cos^{2}{\left(3 x \right)}\right) \sin{\left(3 x \right)}\, dx \]trig-identityUse the identity sin(theta)^2 = 1 - cos(theta)^2.✓ Proved
- \[ = - \int \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}\, dx + \int \sin{\left(3 x \right)}\, dx \]linearity algebraDistribute the sin(3*x) term. The expression is already split.✓ Proved
- \[ = \frac{\cos^{3}{\left(3 x \right)}}{9} + \int \sin{\left(3 x \right)}\, dx \]substitutionIntegrate the second term using substitution u = cos(3*x).✓ Proved
- \[ = \frac{\cos^{3}{\left(3 x \right)}}{9} - \frac{\cos{\left(3 x \right)}}{3} \]antiderivativeEvaluate the integrals.✓ Proved
Answer \( \frac{\left(\cos{\left(6 x \right)} - 5\right) \cos{\left(3 x \right)}}{18} + C \)
✓ 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 |
| 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 trigonometric identities, linearity, and substitution to find the antiderivative. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies trigonometric identities, linearity, and substitution to find the antiderivative. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies trigonometric identities, linearity, and substitution. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (style) 2026-10-17 — Step 4 is a no-op that violates the 'one change' rule by applying no rule; it should be removed or merged. Step 5 incorrectly labels the integration of the second term as 'substitution' when it is an 'antiderivative' step (the substitution logic is implicit in the result, but the step itself is evaluating the integral, not performing the substitution variable change explicitly).gpt-oss:20b: pass 2026-10-17
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 2026-10-19 with SymPy 1.14.0.