Final answer:
To evaluate the integral ∫ 2 (6t i − t² j + 2t³ k) dt, factor out the constant 2, integrate each vector component with respect to t, and then multiply each by 2 to obtain the final result, which is the integrated vector function.
Step-by-step explanation:
The student has asked to evaluate the integral of a vector function with respect to the variable t. Specifically, the integral is ∫ 2 (6t i − t² j + 2t³ k) dt. To evaluate this integral, we can treat each component of the vector function separately and integrate term by term, factor out the constant 2, and then apply the power rule for integration to each term.
Step-by-Step Solution
Factor out the constant from the integral: 2 ∫ (6t i − t² j + 2t³ k) dt.
Integrate each component with respect to t:
Multiply each result by 2:
Combining these results gives us the final integrated vector function: 6t²i − (2/3)t³j + t≤k, where the variable C representing the constant of integration is omitted because it is not provided in the question.