The integral of a vector is simply the vector of the integrals of its components. So, we have:
$$\int_{0}^{1}(8 t \mathbf{i}+t \mathbf{j}-\mathbf{k}) d t = \int_{0}^{1} 8t dt \mathbf{i} + \int_{0}^{1} t dt \mathbf{j} - \int_{0}^{1} dt \mathbf{k}$$
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