Answer
$$-3 \mathbf{i}+(4 \sqrt{2}-2) \mathbf{j}+2 \mathbf{k}$$
Work Step by Step
Given $$ \int_{1}^{2}\left[(6-6 t) \mathbf{i}+3 \sqrt{t} \mathbf{j}+\left(\frac{4}{t^{2}}\right) \mathbf{k}\right] d t $$
Then
\begin{align*}
\int_{1}^{2}\left[(6-6 t) \mathbf{i}+3 \sqrt{t} \mathbf{j}+\left(\frac{4}{t^{2}}\right) \mathbf{k}\right] d t&=\left(6 t-3 t^{2}\right)\bigg|_{1}^{2} \mathbf{i}+ 2 t^{3 / 2} \bigg|_{1}^{2} \mathbf{j}- 4 t^{-1} \bigg|_{1}^{2} \mathbf{k}\\
&=-3 \mathbf{i}+(4 \sqrt{2}-2) \mathbf{j}+2 \mathbf{k}
\end{align*}
