Final Answer:
The indefinite integral of ∫[4e⁴ᵗi+3sin(3t)j+10cos(5t)k]dt is [4e⁴ᵗi - cos(3t)j - 2sin(5t)k] + C, where C is the constant of integration.
Step-by-step explanation:
To find the indefinite integral of the given expression, we integrate each component separately with respect to t. Let's break down the integral:
∫4e⁴ᵗdt results in 4e⁴ᵗ.
∫3sin(3t)dt leads to -cos(3t).
∫10cos(5t)dt gives us -2sin(5t).
Combining these results, the final indefinite integral is [4e⁴ᵗi - cos(3t)j - 2sin(5t)k] + C, where C is the constant of integration. This is because when integrating, we add a constant to the result since the derivative of a constant is zero.
In summary, the integral involves finding the antiderivative for each term, and the final answer includes these antiderivatives along with the constant of integration. The constant accounts for any constant value that may have been present in the original function but disappears when taking the derivative. Therefore, the indefinite integral of the given expression is [4e⁴ᵗi - cos(3t)j - 2sin(5t)k] + C.