Integral of \( \displaystyle \sin{\left(4 x \right)} \cos{\left(2 x \right)} \)
Problem 4.1477 · medium Video
Find \( \displaystyle \int \sin{\left(4 x \right)} \cos{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sin{\left(4 x \right)} \cos{\left(2 x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(0.5 \sin{\left(2 x \right)} + 0.5 \sin{\left(6 x \right)}\right)\, dx \]trig-identityApply the product-to-sum identity: sin(A)cos(B) = 0.5(sin(A+B) + sin(A-B)).≈ Checked numerically
- \[ = 0.5 \int \sin{\left(2 x \right)}\, dx + 0.5 \int \sin{\left(6 x \right)}\, dx \]linearitySplit the integral into two parts using linearity.✓ Proved
- \[ = - 0.25 \cos{\left(2 x \right)} - 0.0833333333333333 \cos{\left(6 x \right)} \]antiderivativeIntegrate each term separately.✓ Proved
- \[ = - \frac{\cos{\left(2 x \right)}}{4} - \frac{\cos{\left(6 x \right)}}{12} \]simplifySimplify the coefficients.✓ Proved
Answer \( - \frac{\cos^{3}{\left(2 x \right)}}{3} + C \)
Lines: 4 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2.66666666666667*sin(x)**6 - 4.0*sin(x)**4 + 1.5*sin(x)**2 - 0.166666666666667*sin(3*x)**2; numeric agreement only, at 24 of 24 sampled points |
| 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left cos(2*x)**3/3 - cos(2*x)/4 - cos(6*x)/12; 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 product-to-sum identities, linearity, and standard antiderivatives. The final result is algebraically equivalent to the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-09 — The solution correctly applies product-to-sum identities, linearity, and standard antiderivatives. The final result is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-11-09qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly applies product-to-sum identities, linearity, and standard antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-08qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly applies product-to-sum identities, linearity, and standard antiderivatives. The final result is algebraically equivalent to the stated answer.gpt-oss:20b: inconclusive 2026-11-08 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final antiderivative is incorrect; the correct result is \n\[-\tfrac{1}{12}\cos(6x)-\tfrac{1}{4}\cos(2x)\n\] which does not simplify to \(-\cos(2x)^3/
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 2026-11-09 with SymPy 1.14.0.