The integral of a vector function is simply the integral of each of its components. So, we can write the integral as follows:
$$\int\left(e^{t} i+\sin t \mathbf{j}+\cos t \mathbf{k}\right) d t = \int e^{t} dt \mathbf{i} + \int \sin t dt \mathbf{j} + \int \cos t dt
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