Final answer:
To find the indefinite integral of the vector-valued function (4t^3i + 8tj - 4tk) with respect to t, integrate each component separately using the power rule, resulting in (t^4i + 4t^2j - 2t^2k) + c, where c is the constant of integration.
Step-by-step explanation:
The question asks us to find the indefinite integral of a vector-valued function. This problem involves calculating the antiderivative of each component of the vector separately. The vector function given is (4t^3i + 8tj - 4tk). To integrate, we apply the power rule of integration to each component, which involves increasing the exponent by one and dividing by the new exponent.
Integrating 4t^3 with respect to t gives us t^4, and we divide by the new exponent 4 to maintain equality. Similarly, we integrate 8t to get 4t^2, and for the -4t, the integral is -2t^2. Each term gets a constant of integration, denoted by c, but we can combine these into one constant because of the linearity of integrals.
The final answer for the indefinite integral is (t^4i + 4t^2j - 2t^2k) + c, where c represents the constant of integration.