The given integral is:
\[
\int_{0}^{1}\left[t e^{2 t} \mathbf{i}+e^{-t} \mathbf{j}+\mathbf{k}\right] d t
\]
This can be broken down into three separate integrals for each component:
\[
\int_{0}^{1} t e^{2t} \, dt \, \mathbf{i}, \quad \int_{0}^{1} e^{-t} \,
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