Evaluate cos^4 3@ sin^3 6@ d@ with limit 0 to pi/6

The given integral is ∫ cos^4(3θ) sin^3(6θ) dθ from 0 to π/6.

Step 1: Use the power-reducing identities to simplify the integral.

cos^4(3θ) = [1 + cos(6θ)]^2 / 4 sin^3(6θ) = (3sin(6θ) - sin^3(6θ)) / 4

Step 2: Substitute these identities into the integral.

∫ [1 + cos(6θ)]^2 * (3sin(6θ) - sin^3(6θ)) / 16 dθ from 0 to π/6

Step 3: Expand the integral and split it into two separate integrals.

= 3/16 ∫ [1 + cos(6θ)]^2 sin(6θ) dθ - 1/16 ∫ [1 + cos(6θ)]^2 sin^3(6θ) dθ from 0 to π/6

Step 4: Evaluate the first integral using the substitution u = 6θ, du = 6dθ.

= 3/16 * 1/6 ∫ [1 + cos(u)]^2 sin(u) du from 0 to π - 1/16 ∫ [1 + cos(6θ)]^2 sin^3(6θ) dθ from 0 to π/6

Step 5: The second integral is more complex and may require numerical methods or special functions to evaluate. However, it is known that the integral of sin^3(θ) from 0 to π is 0, so the second integral is 0.

Therefore, the original integral is equal to 3/16 * 1/6 ∫ [1 + cos(u)]^2 sin(u) du from 0 to π.

Please note that this is a complex integral and may require further simplification or numerical methods to solve.