Final answer:
To evaluate the integral ∫0π/8(cos(2x))4dx, use the trigonometric identity cos^4(2x) = (1/8)(3 + 4cos(4x) + cos(8x)). Split the integral into three parts, evaluate each part separately, and sum the results to obtain the final answer of 3π/64.
Step-by-step explanation:
To evaluate the integral ∫0π/8cos4(2x) dx, we can use a trigonometric identity: cos4(2x) can be expressed as (1/8)(3 + 4cos(4x) + cos(8x)).
Now, we can split the integral into three parts: ∫0π/8(3) dx + ∫0π/8(4cos(4x)) dx + ∫0π/8(cos(8x)) dx.
The first integral evaluates to (3/8)(π/8), the second integral evaluates to 0, and the third integral evaluates to 0 as well. Therefore, the overall integral evaluates to (3/8)(π/8) = 3π/64.