Integral of \( \displaystyle \cos^{3}{\left(2 x \right)} \)
Problem 4.1126 · easy Video
Find \( \displaystyle \int \cos^{3}{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \cos^{3}{\left(2 x \right)}\, dx \]integral algebraStart with the integral of the integrand. Rewrite the power as a product.✓ Proved
- \[ = \int \left(\frac{\cos{\left(4 x \right)}}{2} + \frac{1}{2}\right) \cos{\left(2 x \right)}\, dx \]trig-identityUse the power-reduction identity for cos(2x)^2.✓ Proved
- \[ = \int \left(\frac{\cos{\left(2 x \right)} \cos{\left(4 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{2}\right)\, dx \]linearityDistribute the term cos(2*x).✓ Proved
- \[ = \int \left(\frac{3 \cos{\left(2 x \right)}}{4} + \frac{\cos{\left(6 x \right)}}{4}\right)\, dx \]trig-identity algebra simplifyUse the product-to-sum identity for cos(4x)cos(2x). Simplify the expression inside the integral. Combine like terms.≈ Checked numerically
- \[ = \frac{3 \sin{\left(2 x \right)}}{8} + \frac{\sin{\left(6 x \right)}}{24} \]antiderivative simplifyIntegrate each term separately. Simplify the coefficients.✓ Proved
Answer \( - \frac{\sin^{3}{\left(2 x \right)}}{6} + \frac{\sin{\left(2 x \right)}}{2} + C \)
✓ Nihil obstat Lines: 8 proved, 2 checked numerically. 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (1 - cos(2*x))**2*sin(2*x)/6 - 5*sin(2*x)/24 + sin(4*x)/6 - sin(6*x)/24; numeric agreement only, at 24 of 24 sampled points |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left sin(2*x)**3/6 - sin(2*x)/8 + sin(6*x)/24; numeric agreement only, at 24 of 24 sampled points |
| 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 and linearity to expand the integrand, then integrates term-by-term. The final result is algebraically equivalent to the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies trigonometric identities and linearity to expand the integrand, then integrates term-by-term. The final result is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies trigonometric identities and linearity to expand the integrand, then integrates term-by-term. The final result is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies trigonometric identities and linearity to expand the integrand into a sum of cosines, then integrates term-by-term. The final result is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.