Final answer:
The given integral can be evaluated by splitting it into three separate integrals for each component of the vector. The integral of 4cos(9t) is 4/9, the integral of -2sin(t) is -2, and the integral of sin^2(2t) is (π - 4)/8. Combining these results, the final value of the integral is 4/9i - 2j + (π - 4)/8k.
Step-by-step explanation:
To evaluate the given integral, we can split it into three separate integrals, one for each component of the vector. Let's start with the first integral:
∫[0 to π/2] 4cos(9t) dt
We can integrate this using the basic integral rule: ∫cos(x) dx = sin(x). Applying this rule, we have:
∫[0 to π/2] 4cos(9t) dt = 4sin(9t)/9 evaluated from 0 to π/2
Substituting the limits, we get:
4sin(9π/2)/9 - 4sin(0)/9 = 4/9 - 0 = 4/9
Following the same steps for the second and third integrals, we get:
∫[0 to π/2] -2sin(t) dt = 2cos(t) evaluated from 0 to π/2
2cos(π/2) - 2cos(0) = 0 - 2 = -2
∫[0 to π/2] sin^2(2t) dt = ∫[0 to π/2] (1 - cos(4t))/2 dt
= (t/2 - sin(4t)/8) evaluated from 0 to π/2
(π/4 - sin(2)π/2)/2 - (0 - sin(0)/8) = (π/4 - 1)/2 = (π - 4)/8
Adding all the results together, we have:
∫[0 to π/2] [4cos(9t)i - 2sin(t)j + sin^2(2t)k] dt = 4/9i -2j + (π - 4)/8k