Final answer:
To evaluate the integral ∫(2(4t i - t³ j + 4t⁷ k)) dt from 0, integrate each component separately, and then substitute the upper limit into each component to get the final result.
Step-by-step explanation:
To evaluate the integral ∫(2(4t i - t³ j + 4t⁷ k)) dt from 0, we need to integrate each component separately.
The integral of 4t i with respect to t is 2t² i. The integral of -t³ j with respect to t is -1/4 t⁴ j. The integral of 4t⁷ k with respect to t is 1/2 t⁸ k.
Putting it all together, the integral becomes ∫(2t² i - 1/4 t⁴ j + 1/2 t⁸ k) dt from 0.
When evaluating this integral from 0 to an arbitrary time, we substitute the upper limit into each component. So the final result is 2t² i - 1/4 t⁴ j + 1/2 t⁸ k evaluated at the upper limit.